Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Find a matrix \(B\) such that \(A B\) is the identity matrix. Is there more than one correct result? $$ A=\left[\begin{array}{ll} 2 & 1 \\ 5 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix B such that \(AB\) is the identity matrix is \(B = \[\begin{array}{ll}-2 & 1 \\5 & -2 \end{array}\]\). Since the inverse of a matrix is unique, there is only one correct result.

Step 1: Find the determinant

First, calculate the determinant of matrix \(A\). The determinant \(|A|\) of a 2x2 matrix \(\left[\begin{array}{ll}a & b \\c & d\end{array}\right]\) is given by \(|A| = a*d - b*c\). Here, \(a = 2\), \(b = 1\), \(c = 5\), \(d = 2\), so \(|A| = 2*2 - 1*5 = -1\).

Step 2: Find the adjugate

To find the adjugate of matrix \(A\), swap the elements \(a\) and \(d\) and negate the elements \(b\) and \(c\). Here, \(a = 2\), \(b = 1\), \(c = 5\), \(d = 2\), so adj(A) = \[\begin{array}{ll}2 & -1 \\-5 & 2 \end{array}\].

Step 3: Find the inverse

To find the inverse of matrix \(A\), divide the adjugate by the determinant. So, \(A^{-1} = 1/(-1) *\[\begin{array}{ll}2 & -1 \\-5 & 2 \end{array}\] = \[\begin{array}{ll}-2 & 1 \\5 & -2 \end{array}\]\

Step 4: Check result

To check if the calculated inverse is correct, multiply matrix \(A\) with \(A^{-1} = \[\begin{array}{ll}-2 & 1 \\5 & -2 \end{array}\]\). If the result is the identity matrix, then the inverse is correct.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. It's crucial in matrix algebra, particularly when finding an inverse matrix. The determinant provides information about the matrix, such as whether it is invertible or not. For a 2x2 matrix like \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the determinant, denoted as \(|A|\), is calculated using the formula:\[|A| = ad - bc.\]In our example, with \(a = 2, b = 1, c = 5, \)and \(d = 2, \ \)plug these values into the formula: \[ |A| = 2*2 - 1*5 = 4 - 5 = -1. \]A non-zero determinant (as above, which is \(-1\)) proves that the matrix is invertible. If the determinant were zero, the matrix would not have an inverse.

Adjugate Matrix

Once the determinant is known, the next step in finding the inverse of a matrix is calculating the adjugate matrix. The adjugate of a matrix is formed by taking the transpose of its cofactor matrix.For a 2x2 matrix:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the adjugate matrix, denoted as adj(A), is constructed by swapping the diagonal elements and changing the sign of the off-diagonal elements:\[adj(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\]In our specific problem: - Swap \(a(2)\) and \(d(2)\) - Negate \(b(1)\) and \(c(5)\)Thus, we get:\[adj(A) = \begin{bmatrix} 2 & -1 \ -5 & 2 \end{bmatrix}.\] This new matrix will be critical in finding the inverse by adjusting it with the determinant value.

Inverse Matrix

The inverse matrix, often denoted as \(A^{-1}, \)is a matrix that, when multiplied with the original matrix, yields the identity matrix. Finding the inverse is a fundamental concept because it allows you to solve matrix equations among various other applications.The formula for finding the inverse of a 2x2 matrix is straightforward once you have both the determinant and the adjugate matrix:\[A^{-1} = \frac{1}{|A|} \times adj(A). \]Given \(|A| = -1\) and \(adj(A) = \begin{bmatrix} 2 & -1 \ -5 & 2 \end{bmatrix},\)we can find \(A^{-1}\) as follows:\[A^{-1} = \frac{1}{-1} \times \begin{bmatrix} 2 & -1 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} -2 & 1 \ 5 & -2 \end{bmatrix}. \]To ensure the correctness of the inverse, multiply \(A^{-1} \) with \(A. \) If it results in the identity matrix \(I, \) then the inverse is confirmed correct.The identity matrix for a 2x2 matrix is:\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}.\]This calculating method guarantees that you've found the one unique solution for the inverse matrix under normal circumstances, as recognized by the formula's application.