Find the inverse of a 2 × 2 matrix practice questions | General Mathematics (Year 11, VCE Units 1 & 2, Matrices, Inverse matrices and simultaneous equations)

The inverse of a 2 × 2 matrix

A 2 × 2 inverse is defined only when the determinant is non-zero. After finding the determinant, swap the leading-diagonal entries, negate the other two entries and divide every entry by the determinant

A=\begin{bmatrix}a&b\\c&d\end{bmatrix},\quad A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}

Check the determinant If the determinant is zero, no inverse exists \(\displaystyle \det(A)=2\times3-1\times4=2\ne0\)
Swap the leading diagonal Exchange the top-left and bottom-right entries \(\displaystyle \begin{bmatrix}2&1\\4&3\end{bmatrix}\mapsto\begin{bmatrix}3&1\\4&2\end{bmatrix}\)
Negate the other entries Change the signs of the top-right and bottom-left entries in their existing positions \(\displaystyle \begin{bmatrix}3&1\\4&2\end{bmatrix}\mapsto\begin{bmatrix}3&-1\\-4&2\end{bmatrix}\)
Divide every entry Multiply the rearranged matrix by the reciprocal of the determinant \(\displaystyle A^{-1}=\frac{1}{2}\begin{bmatrix}3&-1\\-4&2\end{bmatrix}=\begin{bmatrix}\frac{3}{2}&-\frac{1}{2}\\-2&1\end{bmatrix}\)

Worked example

Find the inverse of the matrix B and give the entries as exact fractions

B=\begin{bmatrix}4&1\\3&2\end{bmatrix}

\(\displaystyle \det(B)=4\times2-1\times3=8-3=5\)The determinant is 5, so the inverse exists
\(\displaystyle \begin{bmatrix}4&1\\3&2\end{bmatrix}\mapsto\begin{bmatrix}2&-1\\-3&4\end{bmatrix}\)Swap the leading-diagonal entries and negate the other two entries
\(\displaystyle B^{-1}=\frac{1}{5}\begin{bmatrix}2&-1\\-3&4\end{bmatrix}=\begin{bmatrix}\frac{2}{5}&-\frac{1}{5}\\-\frac{3}{5}&\frac{4}{5}\end{bmatrix}\)Divide every rearranged entry by 5 and leave the fractions in lowest terms

Common mistakes

Taking entrywise reciprocals

✕\(\displaystyle A=\begin{bmatrix}4&1\\3&2\end{bmatrix}\Rightarrow A^{-1}=\begin{bmatrix}\frac{1}{4}&1\\\frac{1}{3}&\frac{1}{2}\end{bmatrix}\)✓\(\displaystyle A=\begin{bmatrix}4&1\\3&2\end{bmatrix}\Rightarrow A^{-1}=\frac{1}{5}\begin{bmatrix}2&-1\\-3&4\end{bmatrix}\)

A matrix inverse is found with the determinant and the swap-negate rule, not by taking the reciprocal of each entry

Negating the wrong pair

✕\(\displaystyle A^{-1}=\frac{1}{\det(A)}\begin{bmatrix}-a&b\\c&-d\end{bmatrix}\)✓\(\displaystyle A^{-1}=\frac{1}{\det(A)}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\)

The leading-diagonal entries are swapped, while the other two entries are negated in place

Using a zero determinant

✕\(\displaystyle \det(A)=0\Rightarrow A^{-1}=\begin{bmatrix}0&0\\0&0\end{bmatrix}\)✓\(\displaystyle \det(A)=0\Rightarrow A^{-1}\text{ does not exist}\)

Dividing by a zero determinant is undefined, so a singular matrix has no inverse

Difficulty levels

introUse a determinant of 1 or −1 so the inverse entries are whole numbers

standardCalculate an exact inverse with reduced fractions or terminating decimals

challengeHandle a negative determinant, a singular case and a product check for the inverse

examSelect the inverse from options containing reciprocal and wrong-diagonal errors

Questions about this topic

Why must the determinant be non-zero?+

The inverse rule divides by the determinant. Division by zero is undefined, so a zero determinant means that no inverse exists

Which entries are swapped in the inverse rule?+

The top-left and bottom-right entries are swapped. The top-right and bottom-left entries stay in their positions and have their signs changed

How can an inverse be checked?+

Multiply the matrix by its proposed inverse. A correct inverse gives the identity matrix, with 1s on the leading diagonal and 0s elsewhere

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices

In this chapter

State the order of a matrix and locate its elements 4A Matrix types and order: Identify row, column, square, symmetric and zero matrices Read information from a labelled data matrix 4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matrices Add and subtract matrices 4B Matrix addition and subtraction: Find unknown entries in a matrix equation Multiply a matrix by a scalar Combine scalar multiples of matrices Decide whether a matrix product is defined and give its order Multiply matrices 4D Matrix products and powers: Evaluate powers of a square matrix Identify and construct the identity matrix 4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equation Find the determinant of a 2 × 2 matrix Find the inverse of a 2 × 2 matrix Write simultaneous equations as a matrix equation Solve simultaneous equations using the inverse matrix 4F Matrices for networks: Evaluate powers of a square matrix Represent a network in a matrix Identify the network a connection matrix describes Interpret connection sums and symmetry Count routes using powers of a connection matrix Set up a transition matrix Use a transition matrix to predict the next step 4H State matrices and recurrence: Evaluate powers of a square matrix Generate state matrices by recursion Determine the long-run state of a transition model Use matrix multiplication to answer an application question Use row and column matrices of ones to extract totals