Problem 45 - Precalculus

Question

determine whether \(B\) is the multiplicative inverse of \(A\) using \(A A^{-1}=I\) $$A=\left[\begin{array}{rr}8 & -11 \\\\-5 & 7\end{array}\right] \quad B=\left[\begin{array}{ll} 7 & 11 \\\5 & 8\end{array}\right]$$

Step-by-Step Solution

Answer

Yes, \(B\) is the multiplicative inverse of \(A\) because both \(AB = I\) and \(BA = I\).

Step 1: Understand the Problem

We need to determine if matrix \(B\) is the inverse of matrix \(A\). According to the matrix multiplicative inverse property, if \(B\) is the inverse of \(A\), then \(AB = BA = I\), where \(I\) is the identity matrix of the same dimension, in this case, a 2x2 identity matrix.

Step 2: Calculate AB

To find \(AB\), multiply matrix \(A\) with matrix \(B\):\[ AB = \left[\begin{array}{rr}8 & -11 \ -5 & 7\end{array}\right] \left[\begin{array}{ll} 7 & 11 \ 5 & 8\end{array}\right] \]Perform the matrix multiplication:- First row, first column: \((8 \times 7) + (-11 \times 5) = 56 - 55 = 1\)- First row, second column: \((8 \times 11) + (-11 \times 8) = 88 - 88 = 0\)- Second row, first column: \((-5 \times 7) + (7 \times 5) = -35 + 35 = 0\)- Second row, second column: \((-5 \times 11) + (7 \times 8) = -55 + 56 = 1\)This results in:\[ AB = \left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right] \]

Step 3: Calculate BA

Now let's calculate \(BA\) by multiplying matrix \(B\) with matrix \(A\):\[ BA = \left[\begin{array}{ll} 7 & 11 \ 5 & 8\end{array}\right] \left[\begin{array}{rr}8 & -11 \ -5 & 7\end{array}\right] \]Perform the matrix multiplication:- First row, first column: \((7 \times 8) + (11 \times -5) = 56 - 55 = 1\)- First row, second column: \((7 \times -11) + (11 \times 7) = -77 + 77 = 0\)- Second row, first column: \((5 \times 8) + (8 \times -5) = 40 - 40 = 0\)- Second row, second column: \((5 \times -11) + (8 \times 7) = -55 + 56 = 1\)This results in:\[ BA = \left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right] \]

Step 4: Conclusion

Since \(AB = I\) and \(BA = I\), we have shown that multiplying \(A\) with \(B\) in both orders results in the identity matrix \(I\). Therefore, \(B\) is indeed the multiplicative inverse of \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, involving the multiplication of two matrices to produce a new matrix. Unlike arithmetic multiplication, matrix multiplication follows specific rules that apply to rows and columns. To multiply two matrices, you need to ensure that the number of columns in the first matrix corresponds to the number of rows in the second matrix. For our matrices, which are both 2x2, this condition is naturally met.
When multiplying two matrices, each element of the resulting matrix is the sum of products, obtained by multiplying the corresponding row elements of the first matrix with the corresponding column elements of the second matrix. In the given exercise, you can see how each element of matrix AB is calculated:

• First Row, First Column: Multiply and sum across the row of the first and the column of the second matrix, giving \((8 \times 7) + (-11 \times 5) = 56 - 55 = 1\).
• First Row, Second Column: Similarly, multiply and sum: \((8 \times 11) + (-11 \times 8) = 88 - 88 = 0\).
• Follow a similar process for the second row of A.

This computes the entire product matrix, confirming the multiplicative patterns.​

Identity Matrix

The identity matrix plays a crucial role in matrix operations much like the number 1 does in regular arithmetic. Symbolized as \(I\), the identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. Consider it as the matrix equivalent of the number 1 in multiplication. When a matrix is multiplied by the identity matrix, it remains unchanged. This property is pivotal when verifying the inverse of a matrix.
In the problem, the product matrices AB and BA both equal the 2x2 identity matrix:

• \[ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \] represents the 2x2 identity matrix.
• An important characteristic to note is that when matrix A is multiplied by a correctly determined inverse, the result is this identity matrix \(I\).
• It's the appearance of this result in both orders (AB and BA) that confirms the inverse relationship.

Understanding the identity matrix helps unravel many concepts in linear algebra, making calculations cleaner and comprehensible.

2x2 Matrices

2x2 matrices are one of the simplest forms of matrices, containing just two rows and two columns. Although small, they hold immense importance in mathematics, particularly in transformations and solving systems of linear equations.
The standard form of a 2x2 matrix is:

• \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] where \(a, b, c, \) and \(d\) are elements of the matrix.

In our exercise, both matrices A and B you examine possess this format. One particularly interesting property of 2x2 matrices is how we find their inverses. To determine if B is the inverse of A, as seen, we perform the multiplication check:

• Multiply matrix A by B and verify if the result is the identity matrix.
• Another quick check involves using the determinant. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) to have an inverse, its determinant \(ad-bc\) should not be zero.

These checks confirm inverses, speeding up calculations and confirming the steps employed in the problem.