Problem 9 - College Algebra with Corequisite Support

Question

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$ A=\left[\begin{array}{cc} -2 & \frac{1}{2} \\ 3 & -1 \end{array}\right], \quad B=\left[\begin{array}{ll} -2 & -1 \\ -6 & -4 \end{array}\right] $$

Step-by-Step Solution

Answer

A is the inverse of B since A \times B and B \times A both equal the identity matrix.

Step 1: Recall the Definition of Matrix Inverse

The definition of an inverse matrix states that for two matrices \(A\) and \(B\), \(A\) is the inverse of \(B\) if and only if \(A \times B = I\) and \(B \times A = I\), where \(I\) is the identity matrix.

Step 2: Multiply Matrices A and B

First, compute \(A \times B\). Use the formula for matrix multiplication:\[\left[\begin{array}{cc} -2 & \frac{1}{2} \ 3 & -1 \end{array}\right] \times \left[\begin{array}{ll} -2 & -1 \ -6 & -4 \end{array}\right]\]Calculate each element of the resulting matrix step-by-step:- First row, first column: \((-2 \times -2) + (\frac{1}{2} \times -6) = 4 - 3 = 1\)- First row, second column: \((-2 \times -1) + (\frac{1}{2} \times -4) = 2 - 2 = 0\)- Second row, first column: \((3 \times -2) + (-1 \times -6) = -6 + 6 = 0\)- Second row, second column: \((3 \times -1) + (-1 \times -4) = -3 + 4 = 1\)Thus, \(A \times B = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] = I\).

Step 3: Multiply Matrices B and A

Now compute \(B \times A\). Again, use matrix multiplication:\[\left[\begin{array}{ll} -2 & -1 \ -6 & -4 \end{array}\right] \times \left[\begin{array}{cc} -2 & \frac{1}{2} \ 3 & -1 \end{array}\right]\]Calculate each element of the resulting matrix step-by-step:- First row, first column: \((-2 \times -2) + (-1 \times 3) = 4 - 3 = 1\)- First row, second column: \((-2 \times \frac{1}{2}) + (-1 \times -1) = -1 + 1 = 0\)- Second row, first column: \((-6 \times -2) + (-4 \times 3) = 12 - 12 = 0\)- Second row, second column: \((-6 \times \frac{1}{2}) + (-4 \times -1) = -3 + 4 = 1\)Thus, \(B \times A = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right] = I\).

Step 4: Conclusion

Since both \(A \times B = I\) and \(B \times A = I\), we conclude that matrix \(A\) is indeed the inverse of matrix \(B\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. Unlike regular arithmetic multiplication, matrix multiplication involves combining rows and columns of matrices in a specific manner.
This means that given two matrices, say matrix \( A \) with dimensions \( m \times n \) and matrix \( B \) with dimensions \( n \times p \), the resulting product matrix \( C \) will have dimensions \( m \times p \).

The actual calculation for each element \( c_{ij} \) of matrix \( C \) is performed by taking the dot product of the \( i^{th} \) row of matrix \( A \) and the \( j^{th} \) column of matrix \( B \). Here's how you do it:

• Identify the row from the first matrix and the column from the second matrix.
• Multiply corresponding elements from the row and column.
• Sum these products to get the element \( c_{ij} \) in the resulting matrix.

For example, in our solution, when calculating \((-2 \times -2) + (\frac{1}{2} \times -6)\), you would multiply \(-2\) and \(-2\), then add the product of \(\frac{1}{2}\) and \(-6\), resulting in the first element of the resulting matrix row.

Identity Matrix

The identity matrix is a special kind of matrix known for its simplicity and utility. It acts like the number 1 in ordinary multiplication.
An identity matrix is a square matrix (same number of rows and columns) with ones on its diagonal and zeros elsewhere.

For a matrix \( n \times n \), the identity matrix \( I \) is represented as:

• \( I = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix} \)

It is called the identity matrix because when any matrix is multiplied by the identity matrix, the product is the original matrix. This property is crucial in verifying matrix inverses. In our exercise, matrix multiplication with the identity matrix demonstrated that both \( A \times B \) and \( B \times A \) resulted in identity matrices, proving they are inverses.

Inverse Verification

Verifying the inverse of a matrix involves checking a specific condition using multiplication. For two matrices, \( A \) and \( B \), \( A \) is considered the inverse of \( B \) if both \( A \times B \) and \( B \times A \) result in the identity matrix.
This verification is essential because it confirms that two matrices "undo" each other's operations much like how dividing by a number after multiplying by that same number returns you to the original state.

Steps to verify the inverse:

• Calculate the product \( A \times B \) and confirm if it equals the identity matrix.
• Calculate the product \( B \times A \) and confirm if it also equals the identity matrix.

If both conditions are satisfied, the matrices are verified to be inverses of each other. This process was exactly what was used in the original homework problem to demonstrate that matrices \( A \) and \( B \) were indeed inverses.