Pre-Calculus
Matrices
Inverse matrices
20 practice questions
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Theory + worked examples
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:
\[A^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix},\qquad AA^{-1}=I\]
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.
\[A^{-1}=\dfrac{1}{2}\begin{bmatrix}2&-3\\-2&4\end{bmatrix}=\begin{bmatrix}1&-1.5\\-1&2\end{bmatrix}\]
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.
Example 3 — Verify an inverse
Verify that \(A^{-1}\) from Example 1 satisfies \(AA^{-1}=I\).
Solution
Multiply \(A\) by \(A^{-1}\).
\[\begin{bmatrix}4&3\\2&2\end{bmatrix}\begin{bmatrix}1&-1.5\\-1&2\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]
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.
\[=\begin{bmatrix}\dfrac15&0\\[2pt]0&\dfrac12\end{bmatrix}\]
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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