Problem 32 - 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. (Hint: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{rr}{-10} & {20} \\ {5} & {25}\end{array}\right], B=\left[\begin{array}{rr}{40} & {10} \\ {-20} & {30}\end{array}\right], C=\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1} \\ {1} & {0}\end{array}\right] $$ $$ C A $$

Step-by-Step Solution

Answer

The product \(CA\) is \(\begin{bmatrix} 10 & -20 \\ -5 & -25 \\ -10 & 20 \end{bmatrix}.\)

Step 1: Determine Matrix Dimensions

To perform a matrix multiplication, we need to check the dimensions of the matrices involved. The matrix \(C\) is a \(3 \times 2\) matrix and matrix \(A\) is a \(2 \times 2\) matrix.

Step 2: Check Multiplication Validity

Matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. Here, \(C\) has 2 columns, and \(A\) has 2 rows. Therefore, the multiplication \(CA\) is valid.

Step 3: Multiply Matrices

To multiply matrices \(C\) and \(A\), compute the dot product for each entry. The resulting matrix will have dimensions \(3 \times 2\):Compute each element of the result matrix:- First row, first column: \[(-1)(-10) + (0)(5) = 10\]- First row, second column: \[(-1)(20) + (0)(25) = -20\]- Second row, first column: \[(0)(-10) + (-1)(5) = -5\]- Second row, second column: \[(0)(20) + (-1)(25) = -25\]- Third row, first column: \[(1)(-10) + (0)(5) = -10\]- Third row, second column: \[(1)(20) + (0)(25) = 20\]Thus, the resulting matrix is:\[ \begin{bmatrix} 10 & -20 \ -5 & -25 \ -10 & 20 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

Matrix dimensions are fundamental when it comes to matrix operations such as multiplication. They determine whether such operations are feasible. In our exercise, we have two matrices: matrix \(C\) which is a \(3 \times 2\) matrix and matrix \(A\) which is a \(2 \times 2\) matrix. The dimensions of a matrix are given in terms of the number of rows and columns it contains. Hence, writing a dimension as \(m \times n\), \(m\) represents the rows while \(n\) indicates the columns.

The importance of understanding matrix dimensions comes into play when attempting to multiply matrices. The rule for matrix multiplication necessitates specific alignment of these dimensions, ensuring operations can be carried out smoothly.

Dot Product

The dot product is a critical arithmetic operation used during matrix multiplication. It involves multiplying corresponding elements from a row of the first matrix and a column of the second matrix, then summing these products. In our example, multiplying matrix \(C\) by matrix \(A\) required calculating the dot product for multiple row and column combinations.

Each element in the resulting matrix from multiplying \(C\) and \(A\) corresponds to the dot product of a row from \(C\) and a column from \(A\). For instance, to find the first row, first column of the result, we use:

• (-1 for C)\times(-10 for A) + (0 for C)\times(5 for A)

The dot product not only helps in finding individual elements during multiplication but also gives insight into the alignment and interaction of values between two matrices.

Matrix Operations

Matrix operations like addition, subtraction, and multiplication facilitate many mathematical and real-world applications. Here, the focus is on matrix multiplication. The process involves combining rows and columns through a series of multiplications and additions. The rules governing matrix operations ensure accurate and meaningful results.

For multiplication, ensure the columns of the first matrix match the rows of the second. This operation creates a new matrix—a transformation of the original data structures.

• Addition and subtraction require equal dimensions.
• Multiplication can result in various dimensions, dependent on initial matrix sizes.

Executing matrix operations accurately requires a clear understanding of these basic rules, supporting the efficient handling of linear equations or network models.

Multiplication Validity

Before initiating matrix multiplication, verifying multiplication validity is essential. This verifies that the matrices can be combined through multiplication.

Remember the golden rule: "The number of columns of the first matrix must equal the number of rows of the second matrix." In our given scenario with matrices \(C\) and \(A\):

• Matrix \(C\) has 2 columns.
• Matrix \(A\) has 2 rows.

Here, the dimensions align perfectly, confirming that multiplication is indeed valid. This preemptive check prevents errors and promotes a smoother mathematical process. Additionally, confirming multiplication validity helps predict the dimensions of the resulting matrix, which will have as many rows as the first matrix and as many columns as the second matrix.