The inverse of A undoes what A does. Applying A and then its inverse leaves every vector unchanged.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The definition
A inverse is the matrix satisfying A times its inverse equals the identity, in either order.
The 2 by 2 formula
For rows [a, b] and [c, d], swap a and d, negate b and c, and divide everything by the determinant ad − bc.
When it fails
A determinant of zero means no inverse. The transformation has flattened the plane onto a line, and nothing can unflatten it.
Solving systems
If A is invertible, Ax = b has the single solution x = A inverse times b. Invertibility is exactly the unique-solution case.
The transpose
Transposing flips a matrix across its diagonal, turning rows into columns. Transposing a product also reverses the order.
Undoing a transformation
A⁻¹ reverses what A does, so AA⁻¹ is the identity. It exists only when A is square and the transformation loses no information — that is, when nothing is collapsed.
Solving with the inverse
If A is invertible, Ax = b gives x = A⁻¹b. Conceptually clean, and in practice slower and less stable than elimination, which is what software actually uses.
The inverse of a product reverses order
(AB)⁻¹ = B⁻¹A⁻¹. Undoing a composition means undoing the last step first, exactly as taking off a coat and then a jumper reverses putting them on.
The transpose
Transposing swaps rows and columns. It appears throughout — in dot products written as matrix products, in least squares, and in the definition of symmetric matrices, which have particularly good properties.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(3, 2)
- j-hat lands on(1, 2)
- Determinant4
The shaded parallelogram is the image of the unit square, and its area is 4. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Chen Invert a Matrix
Chen inverts the matrix with rows [4, 7] and [2, 6].
- Step 1
He computes the determinant: 24 − 14 = 10.
Step 4: Your Turn
Practice makes it stick.
The Determinant
Problem 1 of 2
A matrix with rows [3, 1] and [5, 2]. What is its determinant?
The Failure
Problem 2 of 2
A matrix with determinant 0. Does an inverse exist? 1 yes, 0 no.
Undo It
1 of 8
Rows [2, 0] and [0, 5]. What is the determinant?
2 of 8
Rows [1, 2] and [2, 4]. What is the determinant?
3 of 8
A times its inverse gives which matrix? Enter 1 for the identity, 2 for the zero matrix.
4 of 8
Rows [6, 2] and [1, 1]. What is the determinant?
5 of 8
Transposing a 2 by 5 matrix. How many rows does the result have?
6 of 8
A is invertible. How many solutions does Ax = b have?
7 of 8
Sort each matrix by whether it has an inverse.
Tap something to move it.
- Empty
- Empty
8 of 8
The identity matrix. What is its determinant?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Rows [5, 3] and [3, 2]. What is the determinant?
Question 2 of 2
Why does a determinant of zero mean no inverse exists?
What You Learned
- The inverse of A undoes A, and applying both in either order gives the identity.
- For a 2 by 2, swap the diagonal, negate the off-diagonal, and divide by the determinant.
- A determinant of zero means the plane was flattened, so no inverse can exist.