Problem 63 - Precalculus with Limits

Question

In Exercises 63-70, find (a) \(|A|\), (b) \(|B|\), (c) \(AB\), and (d) \(|AB|\). \(A = \left[ \begin{array}{r} -1 && 0 \\ 0 && 3 \end{array} \right]\), \(B = \left[ \begin{array}{r} 2 & 0 \\ 0 & -1 \end{array} \right]\)

Step-by-Step Solution

Answer

The absolute value of A is 3, the absolute value of B is 2, the matrix AB is \(\left[ \begin{array}{r} -2 & 0 \ 0 & -3 \end{array} \right]\), and the absolute value of AB is 6.

Step 1: Find Absolute Value of Matrix A

For the matrix \(A = \left[ \begin{array}{r} -1 && 0 \ 0 && 3 \end{array} \right]\), its absolute value, \(|A|\), is given by \(|-1||3| - |0||0|\) which equals \( |-1 * 3| - 0 = |-3| = 3.\)

Step 2: Find Absolute Value of Matrix B

For the matrix \(B = \left[ \begin{array}{r} 2 & 0 \ 0 & -1 \end{array} \right]\), its absolute value, \(|B|\), is given by \(|2||-1| - |0||0|\) which equals \(|2 * -1| - 0 = |-2| = 2.\)

Step 3: Find Product of Matrices A and B

To get the product AB, each element of the first row of the first matrix is multiplied by the corresponding element of the first column of the second matrix and then they are added to get the first element of the new matrix. Repeating this for all rows of the first matrix and all columns of the second matrix gives the new matrix AB, \(AB = \left[ \begin{array}{r} -1*2 + 0*0 & -1*0 + 0*-1 \ 0*2 + 3*0 & 0*0 + 3*-1 \end{array} \right] = \left[ \begin{array}{r} -2 & 0 \ 0 & -3 \end{array} \right]. \)

Step 4: Find Absolute Value of AB

For the matrix \(AB = \left[ \begin{array}{r} -2 & 0 \ 0 & -3 \end{array} \right]\), its absolute value, \(|AB|\), is given by \(|-2||-3| - |0||0|\) which equals \(|-2 * -3| - 0 = |6| = 6.\)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation where two matrices are combined to produce another matrix. It involves multiplying rows from the first matrix by columns of the second matrix. Each element in the resulting matrix is the sum of all those products. For example, if we have matrices \( A \) and \( B \), the product \( AB \) is not as simple as multiplying individual elements.
It's important to remember the following about matrix multiplication:

• The number of columns in the first matrix must be equal to the number of rows in the second matrix.
• The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
• Matrix multiplication is not commutative, meaning \( AB eq BA \) in general.

In our example, because both \( A \) and \( B \) are 2x2 matrices, they can be successfully multiplied. The product, based on the calculations in the step-by-step solution, results in a matrix \( AB = \left[ \begin{array}{r} -2 & 0 \ 0 & -3 \end{array} \right] \) as every element is derived from these rules.

Absolute Value of a Matrix

The absolute value, or determinant, of a matrix provides a scalar value that has several applications in linear algebra, such as solving systems of linear equations, evaluating matrix invertibility, and calculating volume distortion during linear transformations.
For a 2x2 matrix \( \left[ \begin{array}{r} a & b \ c & d \end{array} \right] \), the determinant is calculated as \( |A| = ad - bc \). This formula gives us insight into the matrix's properties:

• If the determinant is zero, the matrix is singular, meaning it doesn’t have an inverse.
• Otherwise, if non-zero, the matrix is invertible.

In our exercise, we calculate the determinants (absolute values) for matrices \( A \), \( B \), and \( AB \). For matrix \( A \), the determinant is \( |-1| \times |3| - |0| \times |0| = 3 \).
Similarly, for \( B \), \( |2| \times |-1| - |0| \times |0| = 2 \), and for \( AB \), \( |-2| \times |-3| = 6 \). These non-zero results imply that each matrix is invertible.

Diagonal Matrices

A diagonal matrix is a special type of matrix where only the diagonal elements (from the top left corner to the bottom right) can be non-zero, and all other elements are zero. This trait makes them particularly easy to work with when it comes to both multiplication and finding determinants.
Some properties of diagonal matrices include:

• Multiplying two diagonal matrices results in another diagonal matrix.
• They are easy to invert, assuming none of the diagonal elements are zero.
• The determinant of a diagonal matrix is the product of its diagonal elements.

In our case, both matrices \( A \) and \( B \) are diagonal matrices, making the determination of their product straightforward, as seen in the resulting matrix \( AB = \left[ \begin{array}{r} -2 & 0 \ 0 & -3 \end{array} \right] \). Each diagonal element is simply the product of the respective diagonal elements of \( A \) and \( B \), reflecting the convenience of handling diagonal matrices in mathematical computations.