Problem 44 - College Algebra

Question

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ A(C B) $$

Step-by-Step Solution

Answer

The answer to \(A(CB)\) is a matrix: \(\left[\begin{array}{rr} 20 & -4 \ -5 & 15 \ 5 & -1 \end{array}\right]\)

Step 1: Multiplying Matrices C and B

The matrix multiplication of \(C\) and \(B\) can be performed as the number of columns in \(C\) is equal to the number of rows in \(B\). Perform the multiplication as follows: \(CB= \left[\begin{array}{rr} 5 & -1 \ -2 & 2\end{array}\right]\)

Step 2: Multiplying Matrices A and (CB)

Now that we have the result of \(CB\), we can use it in the multiplication of \(A(CB)\). However, the number of columns in \(A\) must equal the number of rows in \((CB)\). Since the matrix \(A\) has two columns and the matrix \((CB)\) has two rows, multiplication can be performed. The resulting matrix will be \(ACB = \left[\begin{array}{rr} 20 & -4 \ -5 & 15 \end{array}\right], \quad\left[\begin{array}{rr} 5 & -1\end{array}\right]\)

Step 3: Final Answer

The resultant matrix of \(A(CB)\) after the multiplication is \(\left[\begin{array}{rr} 20 & -4 \ -5 & 15 \ 5 & -1 \end{array}\right]\)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, involving the combination of two matrices to produce a new matrix. Unlike arithmetic multiplication, matrix multiplication is not performed element-wise. Instead, it's calculated by taking the dot product of the rows and columns from the respective matrices.
To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Here's how it works:

• Select a row from the first matrix and a column from the second matrix.
• Multiply corresponding elements from the row and column together, then sum these products.
• Place the calculated sum into the new matrix, at the row and column position that you considered.
• Repeat the same process for all rows of the first matrix and all columns of the second matrix.

In the example exercise, matrix multiplication is required twice: first to calculate \(CB\), and then to use this result for \(A(CB)\). Understanding these basic mechanics is crucial for executing the process correctly.

Matrix Dimensions

Matrix dimensions refer to the size of a matrix and are expressed in terms of number of rows by number of columns, denoted as \(m \times n\). When performing matrix operations like multiplication, understanding matrix dimensions ensures that operations are possible and helps predict the size of the resultant matrix.
When checking compatibility for matrix multiplication, always verify:

• The columns of the first matrix match the rows of the second matrix, i.e., if \(A\) is \(3 \times 2\), and \(B\) is \(2 \times 2\), multiplication is valid.
• If these conditions hold true, the resulting matrix will have a dimension based on the number of rows from the first matrix and columns from the second matrix.

In the given exercise, matrix \(A\) has dimensions \(3 \times 2\), \(C\) is \(2 \times 2\), and \(B\) is \(2 \times 2\). The operations \(CB\) and then \(A(CB)\) are valid due to these aligned dimensions, allowing for a final product matrix.

Resultant Matrix

The resultant matrix is the product of a matrix multiplication process. Understanding this concept helps to determine the dimension and contents of the output matrix, providing insight into what data is represented post-multiplication.
For any multiplicative operation, the resultant matrix reflects the combined influence of both input matrices. Its dimensions are determined by the rows of the first matrix and the columns of the last one in the operation sequence. Below are some key points to consider:

• The resultant matrix from multiplying a \(3 \times 2\) matrix by a \(2 \times 2\) matrix will be \(3 \times 2\).
• Each entry is a result of summing products across rows from the first matrix and columns from the second, capturing interactions between different elements.

In the exercise, after performing \(A(CB)\), the output is a matrix with dimensions \(3 \times 2\). This resultant matrix encapsulates the effects of both matrix \(A\) and the combined result of \(CB\). Understanding this helps visualize how data is transformed through matrix operations.