91Ó°ÊÓ

To find the determinant of a matrix is to unravel one of the most essential characteristics of that matrix.

In the realm of mathematics, the determinant helps us understand whether a matrix has an inverse or not. For a simple \(2 \times 2\) matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula:

• \( ad - bc \)

This operation results in a single value.

For the given exercise, the matrix is \( \begin{bmatrix} 1 & -2 \ -3 & 2 \end{bmatrix} \). Let's apply the formula:

• The product of the diagonals: \(1 \cdot 2 = 2\)
• Subtract the product of the opposite corners: \((-2) \cdot (-3) = 6\)

Therefore, the determinant is \( 2 - 6 = -4 \). This value indicates that the matrix can be inverted.

The inverse of a matrix is akin to the opposite of a number in basic arithmetic.

Just like how the inverse of a number \(x\) is \(1/x\), the inverse of a matrix \(A\) is another matrix which, when multiplied with \(A\), yields the identity matrix. Not all matrices have an inverse, though.

For a matrix to have an inverse, its determinant must be non-zero. From our previous calculation, our matrix's determinant is \(-4\). As it is not zero, an inverse exists.

To calculate this inverse for a \(2 \times 2\) matrix, we use the formula:

• \(A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\)

The steps to find the inverse involve swapping the diagonal elements and changing the signs of the off-diagonal elements, then dividing everything by the determinant.

This provides an effective method for inverting matrices.

The process of inverting a \(2 \times 2\) matrix is straightforward once you know the steps.

After confirming the determinant is not zero, you follow a systematic approach to find the inverse matrix. Recalling our matrix:

• \( \begin{bmatrix} 1 & -2 \ -3 & 2 \end{bmatrix} \)

We determined the determinant to be \(-4\).

We use the inversion formula:

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

Perform the calculations:

• Each entry of \( \begin{bmatrix} 2 & 2 \ 3 & 1 \end{bmatrix} \) is divided by \(-4\)
• This results in \( \begin{bmatrix} -0.5 & -0.5 \ 0.75 & 0.25 \end{bmatrix} \)

Thus, the inverse matrix has been successfully calculated using a combination of algebraic manipulation and arithmetic, completing the \(2 \times 2\) matrix inversion process.