diff --git a/examples/magnify.js b/examples/magnify.js
index 9193b2f1d2..f3822688ec 100644
--- a/examples/magnify.js
+++ b/examples/magnify.js
@@ -3,7 +3,7 @@ import View from '../src/ol/View.js';
 import TileLayer from '../src/ol/layer/Tile.js';
 import {fromLonLat} from '../src/ol/proj.js';
 import BingMaps from '../src/ol/source/BingMaps.js';
-import {getPixelFromPixel} from '../src/ol/render.js';
+import {getRenderPixel} from '../src/ol/render.js';
 
 const key = 'As1HiMj1PvLPlqc_gtM7AqZfBL8ZL3VrjaS3zIb22Uvb9WKhuJObROC-qUpa81U5';
 
@@ -51,8 +51,8 @@ container.addEventListener('mouseout', function() {
 // after rendering the layer, show an oversampled version around the pointer
 imagery.on('postrender', function(event) {
   if (mousePosition) {
-    const pixel = getPixelFromPixel(event, mousePosition);
-    const offset = getPixelFromPixel(event, [mousePosition[0] + radius, mousePosition[1]]);
+    const pixel = getRenderPixel(event, mousePosition);
+    const offset = getRenderPixel(event, [mousePosition[0] + radius, mousePosition[1]]);
     const half = Math.sqrt(Math.pow(offset[0] - pixel[0], 2) + Math.pow(offset[1] - pixel[1], 2));
     const context = event.context;
     const centerX = pixel[0];
diff --git a/src/ol/render.js b/src/ol/render.js
index 0f39bf8803..330e5aafa6 100644
--- a/src/ol/render.js
+++ b/src/ol/render.js
@@ -5,7 +5,6 @@ import {DEVICE_PIXEL_RATIO} from './has.js';
 import {
   apply as applyTransform,
   create as createTransform,
-  invert as invertTransform,
   multiply as multiplyTransform,
   scale as scaleTransform
 } from './transform.js';
@@ -92,22 +91,22 @@ export function toContext(context, opt_options) {
  */
 export function getVectorContext(event) {
   const frameState = event.frameState;
-  const transform = multiplyTransform(invertTransform(event.pixelTransform.slice()), frameState.coordinateToPixelTransform);
+  const transform = multiplyTransform(event.inversePixelTransform.slice(), frameState.coordinateToPixelTransform);
   return new CanvasImmediateRenderer(
     event.context, frameState.pixelRatio, frameState.extent, transform,
     frameState.viewState.rotation);
 }
 
 /**
- * Gets the pixel of the event's canvas context from the map viewport's css pixel
+ * Gets the pixel of the event's canvas context from the map viewport's CSS pixel.
  * @param {import("./render/Event.js").default} event Render event.
- * @param {import("./pixel.js").Pixel} pixel Css pixel relative to the top-left
+ * @param {import("./pixel.js").Pixel} pixel CSS pixel relative to the top-left
  * corner of the map viewport.
  * @returns {import("./pixel.js").Pixel} Pixel on the event's canvas context.
  * @api
  */
-export function getPixelFromPixel(event, pixel) {
+export function getRenderPixel(event, pixel) {
   const result = pixel.slice(0);
-  applyTransform(invertTransform(event.pixelTransform.slice()), result);
+  applyTransform(event.inversePixelTransform.slice(), result);
   return result;
 }
diff --git a/src/ol/render/Event.js b/src/ol/render/Event.js
index 28caf9fdc2..36d09cf465 100644
--- a/src/ol/render/Event.js
+++ b/src/ol/render/Event.js
@@ -8,22 +8,23 @@ class RenderEvent extends Event {
 
   /**
    * @param {import("./EventType.js").default} type Type.
-   * @param {import("../transform.js").Transform=} opt_pixelTransform Transform.
+   * @param {import("../transform.js").Transform=} opt_inversePixelTransform Transform for
+   *     CSS pixels to rendered pixels.
    * @param {import("../PluggableMap.js").FrameState=} opt_frameState Frame state.
    * @param {?CanvasRenderingContext2D=} opt_context Context.
    * @param {?import("../webgl/Helper.js").default=} opt_glContext WebGL Context.
    */
-  constructor(type, opt_pixelTransform, opt_frameState, opt_context, opt_glContext) {
+  constructor(type, opt_inversePixelTransform, opt_frameState, opt_context, opt_glContext) {
 
     super(type);
 
     /**
-     * Transform from css pixels (relative to the top-left corner of the map viewport)
-     * to render pixel on this event's `context`.
+     * Transform from CSS pixels (relative to the top-left corner of the map viewport)
+     * to rendered pixels on this event's `context`.
      * @type {import("../transform.js").Transform|undefined}
      * @api
      */
-    this.pixelTransform = opt_pixelTransform;
+    this.inversePixelTransform = opt_inversePixelTransform;
 
     /**
      * An object representing the current render frame state.
diff --git a/src/ol/renderer/Map.js b/src/ol/renderer/Map.js
index 281d1eb070..dcd4f96a69 100644
--- a/src/ol/renderer/Map.js
+++ b/src/ol/renderer/Map.js
@@ -9,7 +9,7 @@ import {getWidth} from '../extent.js';
 import {TRUE} from '../functions.js';
 import {visibleAtResolution} from '../layer/Layer.js';
 import {shared as iconImageCache} from '../style/IconImageCache.js';
-import {compose as composeTransform, invert as invertTransform, setFromArray as transformSetFromArray} from '../transform.js';
+import {compose as composeTransform, makeInverse} from '../transform.js';
 
 /**
  * @abstract
@@ -66,8 +66,7 @@ class MapRenderer extends Disposable {
       -viewState.rotation,
       -viewState.center[0], -viewState.center[1]);
 
-    invertTransform(
-      transformSetFromArray(pixelToCoordinateTransform, coordinateToPixelTransform));
+    makeInverse(pixelToCoordinateTransform, coordinateToPixelTransform);
   }
 
   /**
diff --git a/src/ol/renderer/canvas/ImageLayer.js b/src/ol/renderer/canvas/ImageLayer.js
index 405dd4ec05..4c4143a361 100644
--- a/src/ol/renderer/canvas/ImageLayer.js
+++ b/src/ol/renderer/canvas/ImageLayer.js
@@ -6,7 +6,7 @@ import ViewHint from '../../ViewHint.js';
 import {containsExtent, intersects} from '../../extent.js';
 import {getIntersection, isEmpty} from '../../extent.js';
 import CanvasLayerRenderer from './Layer.js';
-import {compose as composeTransform, toString as transformToString} from '../../transform.js';
+import {compose as composeTransform, makeInverse, toString as transformToString} from '../../transform.js';
 
 /**
  * @classdesc
@@ -93,16 +93,17 @@ class CanvasImageLayerRenderer extends CanvasLayerRenderer {
       width = height = size;
     }
 
-    const context = this.context;
-    const canvas = context.canvas;
-
-    const pixelTransform = composeTransform(this.pixelTransform_,
+    // set forward and inverse pixel transforms
+    composeTransform(this.pixelTransform_,
       frameState.size[0] / 2, frameState.size[1] / 2,
       1 / pixelRatio, 1 / pixelRatio,
       rotation,
       -width / 2, -height / 2
     );
-    const canvasTransform = transformToString(pixelTransform);
+    makeInverse(this.inversePixelTransform_, this.pixelTransform_);
+
+    const context = this.context;
+    const canvas = context.canvas;
 
     if (canvas.width != width || canvas.height != height) {
       canvas.width = width;
@@ -150,6 +151,7 @@ class CanvasImageLayerRenderer extends CanvasLayerRenderer {
       canvas.style.opacity = opacity;
     }
 
+    const canvasTransform = transformToString(this.pixelTransform_);
     if (canvasTransform !== canvas.style.transform) {
       canvas.style.transform = canvasTransform;
     }
diff --git a/src/ol/renderer/canvas/Layer.js b/src/ol/renderer/canvas/Layer.js
index 212ee2d995..6417f7807a 100644
--- a/src/ol/renderer/canvas/Layer.js
+++ b/src/ol/renderer/canvas/Layer.js
@@ -7,7 +7,7 @@ import RenderEvent from '../../render/Event.js';
 import RenderEventType from '../../render/EventType.js';
 import {rotateAtOffset} from '../../render/canvas.js';
 import LayerRenderer from '../Layer.js';
-import {invert as invertTransform, create as createTransform, apply as applyTransform, compose as composeTransform} from '../../transform.js';
+import {create as createTransform, apply as applyTransform, compose as composeTransform} from '../../transform.js';
 
 /**
  * @abstract
@@ -28,19 +28,29 @@ class CanvasLayerRenderer extends LayerRenderer {
     this.renderedResolution;
 
     /**
-     * A temporary transform.
+     * A temporary transform.  The values in this transform should only be used in a
+     * function that sets the values.
      * @private
      * @type {import("../../transform.js").Transform}
      */
     this.tempTransform_ = createTransform();
 
     /**
-     * The transform for rendered pixels to viewport CSS pixels.
+     * The transform for rendered pixels to viewport CSS pixels.  This transform must
+     * be set when rendering a frame and may be used by other functions after rendering.
      * @private
      * @type {import("../../transform.js").Transform}
      */
     this.pixelTransform_ = createTransform();
 
+    /**
+     * The transform for viewport CSS pixels to rendered pixels.  This transform must
+     * be set when rendering a frame and may be used by other functions after rendering.
+     * @private
+     * @type {import("../../transform.js").Transform}
+     */
+    this.inversePixelTransform_ = createTransform();
+
     /**
      * @protected
      * @type {CanvasRenderingContext2D}
@@ -102,8 +112,7 @@ class CanvasLayerRenderer extends LayerRenderer {
     applyTransform(frameState.coordinateToPixelTransform, bottomRight);
     applyTransform(frameState.coordinateToPixelTransform, bottomLeft);
 
-    const inverted = invertTransform(this.pixelTransform_.slice());
-
+    const inverted = this.inversePixelTransform_;
     applyTransform(inverted, topLeft);
     applyTransform(inverted, topRight);
     applyTransform(inverted, bottomRight);
@@ -122,46 +131,32 @@ class CanvasLayerRenderer extends LayerRenderer {
    * @param {import("../../render/EventType.js").default} type Event type.
    * @param {CanvasRenderingContext2D} context Context.
    * @param {import("../../PluggableMap.js").FrameState} frameState Frame state.
-   * @param {import("../../transform.js").Transform} pixelTransform Transform.
    * @private
    */
-  dispatchComposeEvent_(type, context, frameState, pixelTransform) {
+  dispatchRenderEvent_(type, context, frameState) {
     const layer = this.getLayer();
     if (layer.hasListener(type)) {
-      const composeEvent = new RenderEvent(type, pixelTransform, frameState,
-        context, null);
-      layer.dispatchEvent(composeEvent);
+      const event = new RenderEvent(type, this.inversePixelTransform_, frameState, context, null);
+      layer.dispatchEvent(event);
     }
   }
 
   /**
    * @param {CanvasRenderingContext2D} context Context.
    * @param {import("../../PluggableMap.js").FrameState} frameState Frame state.
-   * @param {import("../../transform.js").Transform=} opt_transform Transform.
    * @protected
    */
-  preRender(context, frameState, opt_transform) {
-    this.dispatchComposeEvent_(RenderEventType.PRERENDER, context, frameState, opt_transform);
+  preRender(context, frameState) {
+    this.dispatchRenderEvent_(RenderEventType.PRERENDER, context, frameState);
   }
 
   /**
    * @param {CanvasRenderingContext2D} context Context.
    * @param {import("../../PluggableMap.js").FrameState} frameState Frame state.
-   * @param {import("../../transform.js").Transform=} opt_transform Transform.
    * @protected
    */
-  postRender(context, frameState, opt_transform) {
-    this.dispatchComposeEvent_(RenderEventType.POSTRENDER, context, frameState, opt_transform);
-  }
-
-  /**
-   * @param {CanvasRenderingContext2D} context Context.
-   * @param {import("../../PluggableMap.js").FrameState} frameState Frame state.
-   * @param {import("../../transform.js").Transform=} opt_transform Transform.
-   * @protected
-   */
-  dispatchRenderEvent(context, frameState, opt_transform) {
-    this.dispatchComposeEvent_(RenderEventType.RENDER, context, frameState, opt_transform);
+  postRender(context, frameState) {
+    this.dispatchRenderEvent_(RenderEventType.POSTRENDER, context, frameState);
   }
 
   /**
@@ -194,7 +189,7 @@ class CanvasLayerRenderer extends LayerRenderer {
    *    returned, and empty array will be returned.
    */
   getDataAtPixel(pixel, frameState, hitTolerance) {
-    const renderPixel = applyTransform(invertTransform(this.pixelTransform_.slice()), pixel.slice());
+    const renderPixel = applyTransform(this.inversePixelTransform_, pixel.slice());
     const context = this.context;
 
     let data;
diff --git a/src/ol/renderer/canvas/TileLayer.js b/src/ol/renderer/canvas/TileLayer.js
index 3c8c56a8a2..8f389cf15b 100644
--- a/src/ol/renderer/canvas/TileLayer.js
+++ b/src/ol/renderer/canvas/TileLayer.js
@@ -6,7 +6,7 @@ import TileRange from '../../TileRange.js';
 import TileState from '../../TileState.js';
 import {createEmpty, getIntersection, getTopLeft} from '../../extent.js';
 import CanvasLayerRenderer from './Layer.js';
-import {compose as composeTransform, toString as transformToString} from '../../transform.js';
+import {compose as composeTransform, makeInverse, toString as transformToString} from '../../transform.js';
 
 /**
  * @classdesc
@@ -222,15 +222,18 @@ class CanvasTileLayerRenderer extends CanvasLayerRenderer {
       }
     }
 
+
     const canvas = context.canvas;
     const canvasScale = tileResolution / frameState.viewState.resolution / tilePixelRatio;
-    const pixelTransform = composeTransform(this.pixelTransform_,
+
+    // set forward and inverse pixel transforms
+    composeTransform(this.pixelTransform_,
       frameState.size[0] / 2, frameState.size[1] / 2,
       canvasScale, canvasScale,
       rotation,
       -width / 2, -height / 2
     );
-    const canvasTransform = transformToString(pixelTransform);
+    makeInverse(this.inversePixelTransform_, this.pixelTransform_);
 
     if (canvas.width != width || canvas.height != height) {
       canvas.width = width;
@@ -243,7 +246,7 @@ class CanvasTileLayerRenderer extends CanvasLayerRenderer {
       this.clipUnrotated(context, frameState, layerState.extent);
     }
 
-    this.preRender(context, frameState, pixelTransform);
+    this.preRender(context, frameState);
 
     this.renderedTiles.length = 0;
     /** @type {Array<number>} */
@@ -291,7 +294,7 @@ class CanvasTileLayerRenderer extends CanvasLayerRenderer {
       projection, extent, z, tileLayer.getPreload());
     this.scheduleExpireCache(frameState, tileSource);
 
-    this.postRender(context, frameState, pixelTransform);
+    this.postRender(context, frameState);
 
     if (layerState.extent) {
       context.restore();
@@ -302,6 +305,7 @@ class CanvasTileLayerRenderer extends CanvasLayerRenderer {
       canvas.style.opacity = opacity;
     }
 
+    const canvasTransform = transformToString(this.pixelTransform_);
     if (canvasTransform !== canvas.style.transform) {
       canvas.style.transform = canvasTransform;
     }
diff --git a/src/ol/renderer/canvas/VectorLayer.js b/src/ol/renderer/canvas/VectorLayer.js
index 78ffb866d4..452c8b82a6 100644
--- a/src/ol/renderer/canvas/VectorLayer.js
+++ b/src/ol/renderer/canvas/VectorLayer.js
@@ -7,13 +7,12 @@ import {listen, unlisten} from '../../events.js';
 import EventType from '../../events/EventType.js';
 import rbush from 'rbush';
 import {buffer, createEmpty, containsExtent, getWidth} from '../../extent.js';
-import RenderEventType from '../../render/EventType.js';
 import {labelCache} from '../../render/canvas.js';
 import CanvasBuilderGroup from '../../render/canvas/BuilderGroup.js';
 import InstructionsGroupExecutor from '../../render/canvas/ExecutorGroup.js';
 import CanvasLayerRenderer from './Layer.js';
 import {defaultOrder as defaultRenderOrder, getTolerance as getRenderTolerance, getSquaredTolerance as getSquaredRenderTolerance, renderFeature} from '../vector.js';
-import {toString as transformToString} from '../../transform.js';
+import {toString as transformToString, makeScale, makeInverse} from '../../transform.js';
 
 /**
  * @classdesc
@@ -105,8 +104,9 @@ class CanvasVectorLayerRenderer extends CanvasLayerRenderer {
 
     const pixelRatio = frameState.pixelRatio;
 
-    // a scale transform (we could add a createScale function in ol/transform)
-    const pixelTransform = [1 / pixelRatio, 0, 0, 1 / pixelRatio, 0, 0];
+    // set forward and inverse pixel transforms
+    makeScale(this.pixelTransform_, 1 / pixelRatio, 1 / pixelRatio);
+    makeInverse(this.inversePixelTransform_, this.pixelTransform_);
 
     // resize and clear
     const width = Math.round(frameState.size[0] * pixelRatio);
@@ -114,7 +114,7 @@ class CanvasVectorLayerRenderer extends CanvasLayerRenderer {
     if (canvas.width != width || canvas.height != height) {
       canvas.width = width;
       canvas.height = height;
-      const canvasTransform = transformToString(pixelTransform);
+      const canvasTransform = transformToString(this.pixelTransform_);
       if (canvas.style.transform !== canvasTransform) {
         canvas.style.transform = canvasTransform;
       }
@@ -122,7 +122,7 @@ class CanvasVectorLayerRenderer extends CanvasLayerRenderer {
       context.clearRect(0, 0, width, height);
     }
 
-    this.preRender(context, frameState, pixelTransform);
+    this.preRender(context, frameState);
 
     const extent = frameState.extent;
     const viewState = frameState.viewState;
@@ -173,15 +173,11 @@ class CanvasVectorLayerRenderer extends CanvasLayerRenderer {
       }
     }
 
-    if (this.getLayer().hasListener(RenderEventType.RENDER)) {
-      this.dispatchRenderEvent(context, frameState, pixelTransform);
-    }
-
     if (clipped) {
       context.restore();
     }
 
-    this.postRender(context, frameState, pixelTransform);
+    this.postRender(context, frameState);
 
     const opacity = layerState.opacity;
     if (opacity !== canvas.style.opacity) {
diff --git a/src/ol/renderer/canvas/VectorTileLayer.js b/src/ol/renderer/canvas/VectorTileLayer.js
index 8d58fbafec..b9a8a28e0d 100644
--- a/src/ol/renderer/canvas/VectorTileLayer.js
+++ b/src/ol/renderer/canvas/VectorTileLayer.js
@@ -22,11 +22,12 @@ import {
   apply as applyTransform,
   create as createTransform,
   compose as composeTransform,
-  invert as invertTransform,
   reset as resetTransform,
   scale as scaleTransform,
   translate as translateTransform,
-  toString as transformToString
+  toString as transformToString,
+  makeScale,
+  makeInverse
 } from '../../transform.js';
 import CanvasExecutorGroup, {replayDeclutter} from '../../render/canvas/ExecutorGroup.js';
 
@@ -98,6 +99,13 @@ class CanvasVectorTileLayerRenderer extends CanvasTileLayerRenderer {
      */
     this.overlayPixelTransform_ = createTransform();
 
+    /**
+     * The transform for viewport CSS pixels to rendered pixels for the overlay canvas.
+     * @private
+     * @type {import("../../transform.js").Transform}
+     */
+    this.inverseOverlayPixelTransform_ = createTransform();
+
     /**
      * Declutter tree.
      * @private
@@ -384,12 +392,14 @@ class CanvasVectorTileLayerRenderer extends CanvasTileLayerRenderer {
     const rotation = frameState.viewState.rotation;
     const size = frameState.size;
 
+    // set forward and inverse pixel transforms
+    makeScale(this.overlayPixelTransform_, 1 / pixelRatio, 1 / pixelRatio);
+    makeInverse(this.inverseOverlayPixelTransform_, this.overlayPixelTransform_);
+
     // resize and clear
     const canvas = context.canvas;
     const width = Math.round(size[0] * pixelRatio);
     const height = Math.round(size[1] * pixelRatio);
-    this.overlayPixelTransform_[0] = 1 / pixelRatio;
-    this.overlayPixelTransform_[3] = 1 / pixelRatio;
     if (canvas.width != width || canvas.height != height) {
       canvas.width = width;
       canvas.height = height;
@@ -544,7 +554,7 @@ class CanvasVectorTileLayerRenderer extends CanvasTileLayerRenderer {
       return data;
     }
 
-    const renderPixel = applyTransform(invertTransform(this.overlayPixelTransform_.slice()), pixel.slice());
+    const renderPixel = applyTransform(this.inverseOverlayPixelTransform_, pixel.slice());
     const context = this.overlayContext_;
 
     try {
diff --git a/src/ol/transform.js b/src/ol/transform.js
index cbb19c3c39..c7511480fb 100644
--- a/src/ol/transform.js
+++ b/src/ol/transform.js
@@ -162,6 +162,16 @@ export function scale(transform, x, y) {
   return multiply(transform, set(tmp_, x, 0, 0, y, 0, 0));
 }
 
+/**
+ * Creates a scale transform.
+ * @param {!Transform} target Transform to overwrite.
+ * @param {number} x Scale factor x.
+ * @param {number} y Scale factor y.
+ * @return {!Transform} The scale transform.
+ */
+export function makeScale(target, x, y) {
+  return set(target, x, 0, 0, y, 0, 0);
+}
 
 /**
  * Applies translation to the given transform.
@@ -203,30 +213,40 @@ export function compose(transform, dx1, dy1, sx, sy, angle, dx2, dy2) {
 
 /**
  * Invert the given transform.
- * @param {!Transform} transform Transform.
- * @return {!Transform} Inverse of the transform.
+ * @param {!Transform} source The source transform to invert.
+ * @return {!Transform} The inverted (source) transform.
  */
-export function invert(transform) {
-  const det = determinant(transform);
-  assert(det !== 0, 32); // Transformation matrix cannot be inverted
-
-  const a = transform[0];
-  const b = transform[1];
-  const c = transform[2];
-  const d = transform[3];
-  const e = transform[4];
-  const f = transform[5];
-
-  transform[0] = d / det;
-  transform[1] = -b / det;
-  transform[2] = -c / det;
-  transform[3] = a / det;
-  transform[4] = (c * f - d * e) / det;
-  transform[5] = -(a * f - b * e) / det;
-
-  return transform;
+export function invert(source) {
+  return makeInverse(source, source);
 }
 
+/**
+ * Invert the given transform.
+ * @param {!Transform} target Transform to be set as the inverse of
+ *     the source transform.
+ * @param {!Transform} source The source transform to invert.
+ * @return {!Transform} The inverted (target) transform.
+ */
+export function makeInverse(target, source) {
+  const det = determinant(source);
+  assert(det !== 0, 32); // Transformation matrix cannot be inverted
+
+  const a = source[0];
+  const b = source[1];
+  const c = source[2];
+  const d = source[3];
+  const e = source[4];
+  const f = source[5];
+
+  target[0] = d / det;
+  target[1] = -b / det;
+  target[2] = -c / det;
+  target[3] = a / det;
+  target[4] = (c * f - d * e) / det;
+  target[5] = -(a * f - b * e) / det;
+
+  return target;
+}
 
 /**
  * Returns the determinant of the given matrix.
diff --git a/test/spec/ol/transform.test.js b/test/spec/ol/transform.test.js
index 13b5d91846..5a8308ad13 100644
--- a/test/spec/ol/transform.test.js
+++ b/test/spec/ol/transform.test.js
@@ -5,10 +5,12 @@ import {
   setFromArray,
   translate,
   scale,
+  makeScale,
   rotate,
   multiply,
   compose,
   invert,
+  makeInverse,
   apply
 } from '../../../src/ol/transform.js';
 
@@ -68,6 +70,20 @@ describe('ol.transform', function() {
     });
   });
 
+  describe('makeScale()', function() {
+    it('creates a scale transform', function() {
+      const target = create();
+      makeScale(target, 2, 3);
+      expect(target).to.eql([2, 0, 0, 3, 0, 0]);
+    });
+
+    it('returns the target', function() {
+      const target = create();
+      const transform = makeScale(target, 2, 3);
+      expect(transform).to.be(target);
+    });
+  });
+
   describe('rotate()', function() {
     it('applies rotation to a transform', function() {
       const transform = create();
@@ -110,13 +126,39 @@ describe('ol.transform', function() {
 
   describe('invert()', function() {
     it('inverts a transform', function() {
-      let transform = [1, 0, 1, 0, 1, 0];
-      expect(function() {
-        invert(transform);
-      }).to.throwException();
-      transform = [1, 1, 1, 2, 2, 0];
+      const transform = [1, 1, 1, 2, 2, 0];
       expect(invert(transform)).to.eql([2, -1, -1, 1, -4, 2]);
-      expect(transform).to.eql([2, -1, -1, 1, -4, 2]);
+    });
+
+    it('throws if the transform cannot be inverted', function() {
+      const indeterminant = [1, 0, 1, 0, 1, 0];
+      expect(function() {
+        invert(indeterminant);
+      }).to.throwException();
+    });
+
+    it('modifies the source', function() {
+      const source = [1, 1, 1, 2, 2, 0];
+      const inverted = invert(source);
+      expect(inverted).to.eql([2, -1, -1, 1, -4, 2]);
+      expect(source).to.be(inverted);
+    });
+  });
+
+  describe('makeInverse()', function() {
+    it('makes the target the inverse of the source', function() {
+      const source = [1, 1, 1, 2, 2, 0];
+      const target = [1, 0, 0, 1, 0, 0];
+      makeInverse(target, source);
+      expect(source).to.eql([1, 1, 1, 2, 2, 0]);
+      expect(target).to.eql([2, -1, -1, 1, -4, 2]);
+    });
+
+    it('returns the target', function() {
+      const source = [1, 1, 1, 2, 2, 0];
+      const target = [1, 0, 0, 1, 0, 0];
+      const inverted = makeInverse(target, source);
+      expect(target).to.be(inverted);
     });
   });