Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{ll} 1 & -2 \\ 2 & -3 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the matrix \(\begin{bmatrix} 1 & -2 \\ 2 & -3 \end{bmatrix}\) is \(\begin{bmatrix} -3 & 2 \\ -2 & 1 \end{bmatrix}\)

Step 1: Calculate the determinant of the given matrix

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is calculated as \(ad - bc\). Plug in the given values, i.e. \(a = 1, b = -2, c = 2, and d = -3\). So, the determinant is \(1*(-3) - (-2)*2 = -3 - (-4) = 1.\)

Step 2: Check if the determinant is not equal to zero

The determinant we got is 1, which is not equal to zero. Therefore, the given matrix has an inverse. If the determinant was zero, the matrix would not have an inverse.

Step 3: Find the adjoint of the given matrix

The adjoint of a 2x2 matrix is the transpose of the cofactor matrix. However, for a 2x2 matrix, this simplifies a bit further. The adjoint is formed by swapping the positions of \(a\) and \(d\), and changing the signs of \(b\) and \(c\). So, the adjoint of the matrix \(\begin{bmatrix} 1 & -2 \\ 2 & -3 \end{bmatrix}\) is \(\begin{bmatrix} -3 & 2 \\ -2 & 1 \end{bmatrix}\).

Step 4: Construct the inverse of the given matrix

The inverse matrix is obtained by multiplying each element of the adjoint matrix by the reciprocal of the determinant. Since the determinant is 1, the inverse matrix is the same as the adjoint matrix, so the inverse of the given matrix is \(\begin{bmatrix} -3 & 2 \\ -2 & 1 \end{bmatrix}\).

Key Concepts

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, it's really simple to find. Let's look at the pattern: if you have a matrix like \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), you calculate the determinant using the formula \( ad - bc \). This means you multiply the top left entry \( a \) with the bottom right entry \( d \), and then subtract the product of the top right entry \( b \) and the bottom left entry \( c \).
For the matrix in our exercise, we can see that:

• \( a = 1 \), \( b = -2 \)
• \( c = 2 \), \( d = -3 \)

Using our formula, the determinant becomes \( 1(-3) - (-2)2 = -3 - (-4) = 1 \).
This tells us a couple of things: since it's not zero, the matrix does have an inverse, as a zero determinant means no inverse exists.

Adjoint of a Matrix

The adjoint of a matrix plays a crucial role in finding the inverse. For a 2x2 matrix, the adjoint is quite straightforward. You start by taking your original matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). To get the adjoint for a 2x2 matrix, you swap the positions of \( a \) and \( d \). Then, you change the signs of \( b \) and \( c \).
So, for our exercise matrix:

• Originally it's \( \begin{bmatrix} 1 & -2 \ 2 & -3 \end{bmatrix} \)
• Swap \( a \) and \( d \): \( -3 \) goes top left, and \( 1 \) goes bottom right
• Change signs of \( b \) and \( c \): \( -2 \) becomes \( 2 \), and \( 2 \) becomes \( -2 \)

This results in the adjoint \( \begin{bmatrix} -3 & 2 \ -2 & 1 \end{bmatrix} \).
It looks new but is essential for finding the inverse, as it helps when combined with the determinant.

2x2 Matrix Inverse

To find the inverse of the matrix, you need both the determinant and the adjoint. It's quite simple once you have both.
The general formula for the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by multiplying each element of the adjoint by the reciprocal of the determinant: \( \frac{1}{ad-bc} \times \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).
For our problem, we've already seen that the determinant is 1. This simplifies the next step because multiplying by 1 doesn't change the numbers.

• The adjoint was \( \begin{bmatrix} -3 & 2 \ -2 & 1 \end{bmatrix} \)
• Since the determinant is 1, the inverse is just the adjoint

So the inverse of \( \begin{bmatrix} 1 & -2 \ 2 & -3 \end{bmatrix} \) is \( \begin{bmatrix} -3 & 2 \ -2 & 1 \end{bmatrix} \).
This matrix, if multiplied by the original, will give you the identity matrix, confirming it’s indeed the inverse.