Problem 52 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. $$ A=\left[\begin{array}{rrr}{-2} & {0} & {9} \\ {1} & {8} & {-3} \\ {0.5} & {4} & {5}\end{array}\right], B=\left[\begin{array}{ccc}{0.5} & {3} & {0} \\\ {-4} & {1} & {6} \\ {8} & {7} & {2}\end{array}\right], C=\left[\begin{array}{ccc}{1} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {1}\end{array}\right] $$ $$ C A $$

Step-by-Step Solution

Answer

The resulting matrix is \(\begin{bmatrix} -1.5 & 4 & 14 \\ 1 & 8 & -3 \\ -1.5 & 4 & 14 \end{bmatrix}\).

Step 1: Determine Matrix Dimensions

First, identify the dimensions of matrices \(C\) and \(A\). Matrix \(C\) is a \(3 \times 3\) matrix and matrix \(A\) is also a \(3 \times 3\) matrix.

Step 2: Check Matrix Multiplication Compatibility

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. Since both \(C\) and \(A\) are \(3 \times 3\) matrices, they can be multiplied.

Step 3: Perform Matrix Multiplication

Multiply rows of matrix \(C\) by columns of matrix \(A\) to get the resulting matrix. Calculate each element of the resulting matrix:1. First row, first column: \(1(-2) + 0(1) + 1(0.5) = -2 + 0 + 0.5 = -1.5\)2. First row, second column: \(1(0) + 0(8) + 1(4) = 0 + 0 + 4 = 4\)3. First row, third column: \(1(9) + 0(-3) + 1(5) = 9 + 0 + 5 = 14\)4. Second row, first column: \(0(-2) + 1(1) + 0(0.5) = 0 + 1 + 0 = 1\)5. Second row, second column: \(0(0) + 1(8) + 0(4) = 0 + 8 + 0 = 8\)6. Second row, third column: \(0(9) + 1(-3) + 0(5) = 0 - 3 + 0 = -3\)7. Third row, first column: \(1(-2) + 0(1) + 1(0.5) = -2 + 0 + 0.5 = -1.5\)8. Third row, second column: \(1(0) + 0(8) + 1(4) = 0 + 0 + 4 = 4\)9. Third row, third column: \(1(9) + 0(-3) + 1(5) = 9 + 0 + 5 = 14\)

Step 4: Write the Resulting Matrix

The resulting matrix from the multiplication \(CA\) is: \[CA = \begin{bmatrix} -1.5 & 4 & 14 \ 1 & 8 & -3 \ -1.5 & 4 & 14 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

When dealing with matrix multiplication, one of the first steps is determining the dimensions of each matrix involved. A matrix's dimensions are defined as "rows by columns," represented as \(m \times n\). For example, matrix \(C\) and \(A\) both in the given problem are 3x3 matrices, meaning they each have 3 rows and 3 columns.
Understanding these dimensions is crucial as it helps determine if two matrices can be multiplied together. If any of the dimensions differ, the entire operation could become invalid when considering multiplication rules. So always start by ensuring that you understand the size of each matrix.

Matrix Compatibility

Compatibility for multiplication between two matrices depends heavily on their dimensions.
For two matrices, say matrix \(X\) of dimensions \(m \times n\) and matrix \(Y\) of dimensions \(p \times q\), they can only be multiplied if the number of columns in the first matrix (\(n\)) matches the number of rows in the second matrix (\(p\)).

• If the conditions are met, you can proceed with multiplication.
• If not, attempting to multiply them would result in an undefined operation in matrix algebra.

In our exercise, since both matrices \(C\) and \(A\) are 3x3, they satisfy the compatibility condition for multiplication. Whenever solving similar problems, checking compatibility early saves time and prevents mistakes.

Matrix Operations

The operation of multiplying two matrices involves a specific method of combining the numbers in the matrices, called matrix multiplication.
Each element of the resulting matrix is calculated as a sum of products. You multiply the elements of the rows of the first matrix with the corresponding elements of the columns of the second matrix and then add those products.

• Start by taking the first row of the first matrix and multiply it with the first column of the second matrix to get the first element of the resulting matrix.
• Continue this process for each element by proceeding through each combination of rows and columns.

It may seem complex at first, but with practice, the pattern of multiplication becomes clearer, and the computations easier. Ensure each element's computation follows this process to maintain accuracy.

Resulting Matrix

After performing matrix multiplication, you end up with what is called the resulting matrix. The dimensions of this matrix are determined by the number of rows from the first matrix and the number of columns from the second matrix.
For example, if matrix \(C\) is multiplied by matrix \(A\) and both are 3x3 matrices, the resulting matrix is also 3x3. This is because the operation produces as many rows as the first matrix and as many columns as the second matrix has.
In this exercise, upon completing the multiplication for all corresponding elements, we obtain:\[CA = \begin{bmatrix} -1.5 & 4 & 14 \ 1 & 8 & -3 \ -1.5 & 4 & 14 \end{bmatrix}\]Be sure to check each element carefully when calculating, as each represents a unique position derived from the combination of a row and a column from the respective original matrices.