Pre-Calculus Matrices

Inverse matrices

Q: What is an inverse matrix?

A matrix that undoes the original: A times its inverse gives the identity.

Q: How do you find a 2x2 inverse?

Swap a and d, negate b and c, and divide by the determinant a d minus b c.

Q: When does a matrix have no inverse?

When its determinant is zero; such a matrix is called singular.

20 practice questions 0 video lessons Theory + worked examples

Inverse Matrices

California Pre-Calculus • Standard N-VM.10 • Matrices

Inverse Matrices is a topic in Matrices in the California Common Core State Standards. It is aligned to Standard N-VM.10, which requires students to find and use the multiplicative inverse of a matrix.

The inverse \(A^{-1}\) satisfies \(AA^{-1}=I\); for a \(2\times 2\) matrix it swaps the diagonal, negates the off-diagonal, and divides by the determinant.

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Theory

The inverse \(A^{-1}\) of a square matrix undoes it: \(AA^{-1}=A^{-1}A=I\). It plays the role of a reciprocal for matrices.

For a \(2\times 2\) matrix there is a direct formula:

\[A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\ \Rightarrow\ A^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}.\]

An inverse exists only if \(\det A\neq 0\). A singular matrix (\(\det=0\)) has no inverse.

The \(2\times 2\) inverse formula.

An inverse exists only when \(\det\neq 0\).

The 2×2 inverse and its defining property:

\[A^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix},\qquad AA^{-1}=I\]

the inverse swaps a and d, negates b and c, and divides by the determinant

Steps: swap the diagonal, negate the off-diagonal, divide by the determinant.

How to find a 2×2 inverse

Compute \(\det=ad-bc\); if it is \(0\), stop — no inverse.
Swap \(a\) and \(d\).
Negate \(b\) and \(c\).
Divide every entry by the determinant, and verify with \(AA^{-1}=I\).

Example 1 — Invert a 2×2

Find the inverse of \(A=\begin{bmatrix}4&3\\2&2\end{bmatrix}\).

Solution

Determinant \(=4(2)-3(2)=2\). Swap \(a,d\), negate \(b,c\), divide by the determinant.

\[A^{-1}=\dfrac{1}{2}\begin{bmatrix}2&-3\\-2&4\end{bmatrix}=\begin{bmatrix}1&-1.5\\-1&2\end{bmatrix}\]

inverse is 1, negative 1.5, negative 1, 2

Example 2 — No inverse

Does \(\begin{bmatrix}2&4\\1&2\end{bmatrix}\) have an inverse?

Solution

Check the determinant.

\(\det\)

\(=\)

\(2(2)-4(1)=0\)

Determinant \(0\) \(\Rightarrow\) no inverse.

determinant is 0, so there is no inverse

Example 3 — Verify an inverse

Verify that \(A^{-1}\) from Example 1 satisfies \(AA^{-1}=I\).

Solution

Multiply \(A\) by \(A^{-1}\).

\[\begin{bmatrix}4&3\\2&2\end{bmatrix}\begin{bmatrix}1&-1.5\\-1&2\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]

the product is the identity, confirming the inverse

Example 4 — Inverse of a diagonal matrix

Find the inverse of \(\begin{bmatrix}5&0\\0&2\end{bmatrix}\).

Solution

For a diagonal matrix, invert each diagonal entry.

\[=\begin{bmatrix}\dfrac15&0\\[2pt]0&\dfrac12\end{bmatrix}\]

inverse is one fifth and one half on the diagonal

Common pitfalls

Check the determinant first. A \(0\) determinant means no inverse exists.

Swap the diagonal, negate the off-diagonal — don't just take reciprocals of entries.

Divide the whole matrix by \(\det\), not just one entry.

Frequently asked questions

What is an inverse matrix?

A matrix \(A^{-1}\) with \(AA^{-1}=A^{-1}A=I\); it undoes the original matrix.

How do you find a 2x2 inverse?

Swap \(a\) and \(d\), negate \(b\) and \(c\), and divide by the determinant \(ad-bc\).

When does a matrix have no inverse?

When its determinant is \(0\); such a matrix is called singular.

How do you check an inverse is correct?

Multiply it by the original; the product should be the identity matrix.

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Solving linear systems with matrices

Inverse matrices

Inverse Matrices

📖 Theory

How to find a 2×2 inverse

Common pitfalls

Frequently asked questions

Theory