Problem 2 - Precalculus

Question

Find the inverse of the matrix or state that the matrix is not invertible. $$ B=\left[\begin{array}{rr} 12 & -7 \\ -5 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of matrix \( B \) is \( B^{-1} = \begin{bmatrix} 3 & 7 \\ 5 & 12 \end{bmatrix} \).

Step 1: Calculate the Determinant of Matrix B

To determine if the matrix is invertible, we first need to calculate its determinant. The determinant of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by the formula \(\text{det}(A) = ad - bc\). For matrix \( B \), \( a = 12 \), \( b = -7 \), \( c = -5 \), and \( d = 3 \). Thus, the determinant \( \text{det}(B) = (12)(3) - (-7)(-5) = 36 - 35 = 1 \). Since the determinant is non-zero, matrix \( B \) is invertible.

Step 2: Use the Formula for the Inverse of a 2x2 Matrix

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) with a non-zero determinant, the inverse is given by \( A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). For matrix \( B \), with \( a = 12 \), \( b = -7 \), \( c = -5 \), and \( d = 3 \), and the determinant previously found to be 1, \( B^{-1} = \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \). Since the determinant was 1, the inverse is simply the adjugate matrix of \( B \).

Step 3: Write the Inverse Matrix

The inverse matrix \( B^{-1} \) is obtained directly from the formula. Therefore, the inverse of matrix \( B \) is given by \( B^{-1} = \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \). This matrix, when multiplied by the original matrix \( B \), should yield the identity matrix, confirming its correctness.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can give us insights into the properties of the matrix. Specifically, for a 2x2 matrix like our matrix \( B \), the determinant tells us if the matrix has an inverse or not. If the determinant is zero, the matrix is said to be "singular" and does not have an inverse. Conversely, if the determinant is not zero, the matrix is "non-singular" and does have an inverse.

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula:

• \( \text{det}(A) = ad - bc \)

In our example, the given matrix \( B \) has elements \( a = 12 \), \( b = -7 \), \( c = -5 \), and \( d = 3 \). Thus, the determinant can be calculated as:

\[ \text{det}(B) = (12)(3) - (-7)(-5) = 36 - 35 = 1 \]

This results in a determinant of \( 1 \), which is a confirmation that the matrix \( B \) is invertible.

Inverse of a 2x2 Matrix

Finding the inverse of a matrix is of great importance in mathematical computations since it is closely associated with solving linear systems. For any 2x2 invertible matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse matrix \( A^{-1} \) can be calculated if the determinant \( ad-bc \) is non-zero.

The formula for finding the inverse of a 2x2 matrix is:

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

Applying this formula to our matrix \( B \) with the determinant calculated as 1, we plug in the values:

\[ B^{-1} = \frac{1}{1}\begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} = \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \]

Notice that since the determinant is 1, the inverse matrix is simply the adjugate of \( B \). The importance of finding this inverse is that multiplying \( B \) by its inverse will result in the identity matrix, thus proving its correctness.

Adjugate Matrix

The adjugate matrix, sometimes called the adjoint, plays a crucial role in finding the inverse of a matrix. It consists of the transpose of the cofactor matrix. For a 2x2 matrix like \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the adjugate matrix is obtained by swapping the diagonal elements and changing the signs of the off-diagonal elements.

More specifically, the adjugate matrix \( \text{adj}(A) \) for a 2x2 matrix is:

• \( \text{adj}(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \)

In our specific example with matrix \( B \), whose entries are \( a = 12 \), \( b = -7 \), \( c = -5 \), and \( d = 3 \), the adjugate matrix is:

\[ \text{adj}(B) = \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \]

This matrix mirrors the structure of \( B \) but with rearrangements and sign changes, and because our matrix had a determinant of \( 1 \), the adjugate matrix was also the inverse matrix of \( B \). This simplifies computations significantly when the determinant is 1.