Inverse of a 2×2 Matrix Calculator

This Inverse of a 2×2 Matrix Calculator helps you find the inverse of a 2×2 matrix and shows each step clearly. It works by first finding the determinant, then swapping the main diagonal entries, changing the signs of the off-diagonal entries, and multiplying by the reciprocal of the determinant. This makes it useful for checking answers, understanding how a matrix inverse is found, and practising linear algebra step by step.

Step-by-step method

1. Identify the entries a, b, c, and d in the 2×2 matrix.
2. Find the determinant using det( A ) = ad − bc, then check that it is not 0.
3. Use the inverse matrix pattern shown in the formula.
4. Multiply the new matrix by 1 / det( A ) and simplify.

Formulas:

Example 1: Take the values below.

Step 1 - Identify the entries a, b, c, and d in the 2×2 matrix.

In this problem: For a 2×2 matrix, a is top-left, b is top-right, c is bottom-left, and d is bottom-right.

Step 2 - Find the determinant using det( A ) = ad − bc, then check that it is not 0.

In this problem: The determinant is \(1\). Since \(1 \neq 0\), the matrix has an inverse.

Step 3 - Use the inverse matrix pattern shown in the formula.

In this problem: Use the inverse matrix pattern from the formula, then substitute this problem’s values.

Step 4 - Multiply the new matrix by 1 / det( A ) and simplify.

In this problem: Multiply each entry by the reciprocal of the determinant and simplify.

Final answer:

Example 2: Take the values below.

Step 1 - Identify the entries a, b, c, and d in the 2×2 matrix.

In this problem: For a 2×2 matrix, a is top-left, b is top-right, c is bottom-left, and d is bottom-right.

Step 2 - Find the determinant using det( A ) = ad − bc, then check that it is not 0.

In this problem: The determinant is \(6\). Since \(6 \neq 0\), the matrix has an inverse.

Step 3 - Use the inverse matrix pattern shown in the formula.

In this problem: Use the inverse matrix pattern from the formula, then substitute this problem’s values.

Step 4 - Multiply the new matrix by 1 / det( A ) and simplify.

In this problem: Multiply each entry by the reciprocal of the determinant and simplify.

Final answer:

See Example 1 Hide Example 1

Problem

\(A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\)

Approach

Step-by-step method

1. Identify the entries a, b, c, and d in the 2×2 matrix.
2. Find the determinant using det( A ) = ad − bc, then check that it is not 0.
3. Use the inverse matrix pattern shown in the formula.
4. Multiply the new matrix by 1 / det( A ) and simplify.

Step 1

Step 1 - Identify the entries a, b, c, and d in the 2×2 matrix.

In this problem: For a 2×2 matrix, a is top-left, b is top-right, c is bottom-left, and d is bottom-right.

\(A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix},\quad a = 2,\; b = 1,\; c = 5,\; d = 3\)

Step 2

Step 2 - Find the determinant using det( A ) = ad − bc, then check that it is not 0.

In this problem: The determinant is \(1\). Since \(1 \neq 0\), the matrix has an inverse.

\(\det(A) = 2 \cdot 3 - 1 \cdot 5 = 6 - 5 = 1,\quad 1 \neq 0\)

Step 3

Step 3 - Use the inverse matrix pattern shown in the formula.

In this problem: Use the inverse matrix pattern from the formula, then substitute this problem’s values.

\(\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}\)

Step 4

Step 4 - Multiply the new matrix by 1 / det( A ) and simplify.

In this problem: Multiply each entry by the reciprocal of the determinant and simplify.

\(A^{-1} = \frac{1}{1} \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}\)

Final Answer

\(A^{-1} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}\)

See Example 2 Hide Example 2

Problem

\(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}\)

Approach

Step-by-step method

1. Identify the entries a, b, c, and d in the 2×2 matrix.
2. Find the determinant using det( A ) = ad − bc, then check that it is not 0.
3. Use the inverse matrix pattern shown in the formula.
4. Multiply the new matrix by 1 / det( A ) and simplify.

Step 1

Step 1 - Identify the entries a, b, c, and d in the 2×2 matrix.

In this problem: For a 2×2 matrix, a is top-left, b is top-right, c is bottom-left, and d is bottom-right.

\(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix},\quad a = 4,\; b = -2,\; c = 1,\; d = 1\)

Step 2

Step 2 - Find the determinant using det( A ) = ad − bc, then check that it is not 0.

In this problem: The determinant is \(6\). Since \(6 \neq 0\), the matrix has an inverse.

\(\det(A) = 4 \cdot 1 - (-2) \cdot 1 = 4 - (-2) = 6,\quad 6 \neq 0\)

Step 3

Step 3 - Use the inverse matrix pattern shown in the formula.

In this problem: Use the inverse matrix pattern from the formula, then substitute this problem’s values.

\(\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ -1 & 4 \end{pmatrix}\)

Step 4

Step 4 - Multiply the new matrix by 1 / det( A ) and simplify.

In this problem: Multiply each entry by the reciprocal of the determinant and simplify.

\(A^{-1} = \frac{1}{6} \begin{pmatrix} 1 & 2 \\ -1 & 4 \end{pmatrix} = \begin{pmatrix} \frac{1}{6} & \frac{1}{3} \\ -\frac{1}{6} & \frac{2}{3} \end{pmatrix}\)

Final Answer

\(A^{-1} = \begin{pmatrix} \frac{1}{6} & \frac{1}{3} \\ -\frac{1}{6} & \frac{2}{3} \end{pmatrix}\)