Problem 48 - Precalculus with Limits

Question

In Exercises 47-52, if possible, find (a) \(AB\), (b) \(BA\), and (c) \(A^2\). (Note: \(A^2 = AA\).) \(A=\left[\begin{array}{r} 6 & 3 \\ -2 & -4 \end{array}\right]\), \(B=\left[\begin{array}{r} -2 && 0 \\ 2 && 4 \end{array}\right]\)

Step-by-Step Solution

Answer

Based on the solution steps above, the three required matrix products are: (a) AB = \left[\begin{array}{cc} 0 & 12 \ 0 & -16 \end{array}\right], (b) BA = \left[\begin{array}{cc} -12 & -6 \ 0 & -8 \end{array}\right], and (c) A^2 = \left[\begin{array}{cc} 32 & 2 \ -8 & 10 \end{array}\right]

Step 1: Calculation of matrix product AB

First, calculate the matrix product AB. This is done by taking the dot product of the rows of matrix A with the columns of matrix B. For the first element of the result matrix, multiply the first element of the first row in matrix A with the first element of the first column in matrix B, add this to the product of the second element of the first row in A and the second element of the first column in B. Repeat this process for all elements in the matrices: \[AB = \left[\begin{array}{cc} (6*-2 + 3*2) & (6*0 + 3*4) \ (-2*-2 + -4*2) & (-2*0 + -4*4) \end{array}\right] = \left[\begin{array}{cc} 0 & 12 \ 0 & -16 \end{array}\right]\]

Step 2: Calculation of matrix product BA

Next, compute the matrix product BA in a similar way. Since the order of multiplication matters for matrices, the result will not necessarily be the same as for AB. \[BA = \left[\begin{array}{cc} (-2*6 + 0*-2) & (-2*3 + 0*-4) \ (2*6 + 4*-2) & (2*3 + 4*-4) \end{array}\right] = \left[\begin{array}{cc} -12 & -6 \ 0 & -8 \end{array}\right]\]

Step 3: Calculation of matrix square A^2

The square of a matrix A, denoted by A^2, is the matrix product AA. So compute A^2 in a similar way to the earlier steps: \[A^2 = \left[\begin{array}{cc} (6*6 + 3*-2) & (6*3 + 3*-4) \ (-2*6 + -4*-2) & (-2*3 + -4*-4) \end{array}\right] = \left[\begin{array}{cc} 32 & 2 \ -8 & 10 \end{array}\right]\]

Key Concepts

Matrix Product

Matrix multiplication, or matrix product, involves multiplying two matrices by calculating the sum of products of their respective elements. This operation is only possible if the number of columns in the first matrix matches the number of rows in the second matrix. To find the product of two matrices, you take each element from a row of the first matrix and multiply it with each corresponding element in a column of the second matrix, then sum these products.
For example, to calculate the matrix product of matrices A and B, as demonstrated in the exercise, one would work through each element:

• For the first element in the result matrix, multiply the first element of the first row of matrix A with the first element of the first column of matrix B, then add the result of multiplying the second element of the first row of matrix A by the second element in the same column of matrix B.
• Repeat the process for each element in the respective matrices to fill the entire result matrix.

This requires attention to detail and a good handle on basic arithmetic operations.

Square of a Matrix

The square of a matrix is denoted as \(A^2\) and involves multiplying the matrix by itself. This concept is straightforward as it involves a matrix multiplied by itself, meaning the operation involves the same processes as regular matrix multiplication.
To compute the square of a matrix, follow these steps:

• Arrange the matrix with itself side-by-side for multiplication.
• Multiply each row of the first matrix with each column of the second matrix, as in standard matrix multiplication, and proceed to fill the resulting matrix with these computations.

By doing so, for matrix A, you'll find its square \(A^2\) is another matrix signifying repeated linear transformations that the original matrix was structured for. The squaring of a matrix often simplifies operations for problems involving iterative processes.

Order of Multiplication

In matrix operations, the order of multiplication significantly influences the result as matrix multiplication is not commutative. This means that generally \(AB eq BA\). The order in which matrices are multiplied can dramatically affect the resulting matrix.
Consider the difference emphasized in the solution:

• Matrix product \(AB\) leads to a different matrix than the product \(BA\) due to the differing orientations of rows and columns for each operation.
• This outcome stresses the importance of correctly understanding the sequence of matrix operations, as reversing the order often yields different results.

This principle is crucial to remember when working with various matrix equations and operations, to ensure accuracy in results.