Problem 43 - 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. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ A(B C) $$

Step-by-Step Solution

Answer

The resulting matrix of the operation \(A(BC)\) is \[\left[\begin{array}{rr} 16 & -16 \ -12 & 12 \ 0 & 0 \end{array}\right]\]

Step 1: Multiply Matrix B and C

First, multiply matrices B and C. Start by taking the first row of matrix B and first column in matrix C, multiply corresponding elements and add them up to get the first element of the resulting matrix. Repeat the process for all rows and columns of matrices B and C. The result for \(BC\) is: \[BC=\left[\begin{array}{rr} 5*1 + 1*-1 & 5*-1 + 1*1 \ -2*1 + -2*-1 & -2*-1 + -2*1 \end{array}\right] = \left[\begin{array}{rr} 4 & -4 \ 0 & 0 \end{array}\right]\]

Step 2: Multiply Matrix A by the result of BC

Next, multiply matrix A with the resulting matrix BC. The size of matrix A is 3x2 and matrix BC is 2x2, thus they can be multiplied. Following the same steps: \[ABC=\left[\begin{array}{rr} 4*4 + 0*0 & 4*-4 + 0*0 \ -3*4 + 5*0 & -3*-4 + 5*0 \ 0*4 + 1*0 & 0*-4 + 1*0 \end{array}\right] = \left[\begin{array}{rr} 16 & -16 \ -12 & 12 \ 0 & 0 \end{array}\right]\]

Key Concepts

Matrix Multiplication

Matrix multiplication is a process in which two matrices are combined to produce a new matrix. Here, each element of the resultant matrix is found by multiplying corresponding elements from the rows of the first matrix and columns of the second matrix, and then summing these products.

• Step-by-Step Procedure: Begin by selecting a row from the first matrix and a column from the second. Multiply the corresponding elements and then add the results.
• Example: For matrices B and C, the entry in the first row and first column of the resulting matrix is determined by multiplying the corresponding elements in the first row of B with those in the first column of C.

Practicing matrix multiplication will enhance your understanding of how matrices interact with one another.

Matrix Dimensions

The dimensions of a matrix are key to understanding whether certain operations like multiplication can be performed. Each dimension describes the number of rows and columns.

• Determining Dimensions: To find the dimensions of a matrix, count the number of rows and columns. For example, matrix A is 3x2, meaning it has 3 rows and 2 columns.
• Compatibility for Multiplication: Matrices can only be multiplied when the number of columns in the first matrix equals the number of rows in the second matrix.

This compatibility check is crucial before performing matrix multiplication to avoid undefined operations.

Matrix Addition

Matrix addition is a simpler operation compared to multiplication. Two matrices can be added together if they share the same dimensions, with each element in the resulting matrix being the sum of the corresponding elements from the two matrices.

• Example: Consider two matrices A and C, both with dimensions 3x2. Their sum is found by adding the elements in the first row of A to those in the first row of C, and continuing this process for the remaining rows.
• Why Important: Understanding addition is fundamental, as it allows for combining matrix equations and enhances comprehension of more complex operations.

While the problem did not specifically involve matrix addition, recognizing this operation forms a foundational element in matrix manipulation.

Matrix Result

The end result of a matrix operation is often a new matrix which represents a transformation or combination of the original matrices.

• Calculation of Result: In our example, the multiplication of matrices B and C yielded a matrix that was then multiplied by matrix A to get the final result of ABC.
• Interpretation: This result encapsulates the combined effects of both multiplications, providing insights into the properties achieved through the operation.

Understanding the resultant matrix is crucial as it often provides the solution to various mathematical problems or practical situations involving multiple matrices.