Find the inverse of the matrix [2 5] [6 -3]
Understand the Problem
The question asks to find the inverse of a provided 2x2 matrix. The general approach is to first calculate the determinant of the matrix, and then use that value to compute the inverse, by swapping the diagonal elements, negating the off-diagonal elements, and dividing by the determinant.
Answer
$\begin{bmatrix} \frac{1}{12} & \frac{5}{36} \\ \frac{1}{6} & -\frac{1}{18} \end{bmatrix}$
Answer for screen readers
$A^{-1} = \begin{bmatrix} \frac{1}{12} & \frac{5}{36} \ \frac{1}{6} & -\frac{1}{18} \end{bmatrix}$
Steps to Solve
- Write down the given matrix
The given matrix $A$ is: $A = \begin{bmatrix} 2 & 5 \ 6 & -3 \end{bmatrix}$
- Calculate the determinant of the matrix
For a 2x2 matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is $ad - bc$. So, for matrix $A$, the determinant is $(2 \times -3) - (5 \times 6) = -6 - 30 = -36$. $\det(A) = -36$
- Find the adjugate (or adjoint) of the matrix
For a 2x2 matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$, the adjugate is $\begin{bmatrix} d & -b \ -c & a \end{bmatrix}$. So, for matrix $A$, the adjugate is $\begin{bmatrix} -3 & -5 \ -6 & 2 \end{bmatrix}$.
- Calculate the inverse of the matrix
The inverse of a matrix $A$ is given by $A^{-1} = \frac{1}{\det(A)} \times \text{adjugate}(A)$. $A^{-1} = \frac{1}{-36} \begin{bmatrix} -3 & -5 \ -6 & 2 \end{bmatrix} = \begin{bmatrix} \frac{-3}{-36} & \frac{-5}{-36} \ \frac{-6}{-36} & \frac{2}{-36} \end{bmatrix} = \begin{bmatrix} \frac{1}{12} & \frac{5}{36} \ \frac{1}{6} & -\frac{1}{18} \end{bmatrix}$
$A^{-1} = \begin{bmatrix} \frac{1}{12} & \frac{5}{36} \ \frac{1}{6} & -\frac{1}{18} \end{bmatrix}$
More Information
The inverse of a matrix, when multiplied by the original matrix, results in the identity matrix. The identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere.
Tips
A common mistake is to incorrectly calculate the determinant, especially with the signs. Another mistake is not swapping the positions of $a$ and $d$ or not changing the signs of $b$ and $c$ when finding the adjugate matrix. Also, remember to multiply every element of the adjugate matrix by the scalar $1/\text{det}(A)$.
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