Problem 40 - 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 result of the matrix operation \(A(B+C)\) is \(\begin{bmatrix} 24 & 0 \\ -18 & -5 \\ 0 & -1 \end{bmatrix}.\)

Step 1: Matrix Addition of B and C

Because matrices B and C have the same dimensions, we can just perform element-wise addition. The element at row i, column j in the result is the sum of the elements at row i, column j in B and C.\nThus, \(B+C=\begin{bmatrix} 5+1 & 1+(-1) \\ -2+(-1) & -2+1 \end{bmatrix}.\) Simplifying each entry, we obtain \(B+C=\begin{bmatrix} 6 & 0 \\ -3 & -1 \end{bmatrix}.\)

Step 2: Matrix Multiplication of A with (B+C)

Matrix A has dimensions 3x2 and the matrix (B+C) has dimensions 2x2. Because the number of columns in A is equal to the number of rows in (B+C), we can perform the multiplication. The element at row i, column j in the result is obtained by taking the dot product of the i-th row of A with the j-th column of (B+C).\nThus, \(A(B+C) = \begin{bmatrix} 4*6 + 0*0 & 4*0 + 0*(-1) \\ -3*6 + 5*0 & -3*0 + 5*(-1) \\ 0*6 + 1*0 & 0*0 + 1*(-1) \end{bmatrix}.\) After doing the operations within each entry, we get the result as \(A(B+C) = \begin{bmatrix} 24 & 0 \\ -18 & -5 \\ 0 & -1 \end{bmatrix}.\)

Key Concepts

Matrix Addition

When it comes to matrix operations, one of the simplest to understand is matrix addition. Matrix addition is quite straightforward: only matrices of the same dimensions can be added. This means the matrices need to have the same number of rows and columns. You combine them by adding corresponding elements. For instance, if you have two matrices, say Matrix B and Matrix C, which both have 2 rows and 2 columns:

• Matrix B: \(\begin{bmatrix} 5 & 1 \ -2 & -2 \end{bmatrix}\)
• Matrix C: \(\begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix}\)

To add these matrices, simply add corresponding elements:

• First row, first column: \(5 + 1 = 6\)
• First row, second column: \(1 + (-1) = 0\)
• Second row, first column: \(-2 + (-1) = -3\)
• Second row, second column: \(-2 + 1 = -1\)

This results in a new matrix: \(B+C = \begin{bmatrix} 6 & 0 \ -3 & -1 \end{bmatrix}\). Matrix addition is foundational in understanding more complex operations like matrix multiplication.

Matrix Multiplication

Matrix multiplication is a bit more involved than matrix addition. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. If this condition is met, you perform what's called the dot product for each element in the resulting matrix. For multiplication of Matrix A and Matrix \((B+C)\), where:

• Matrix A: \(3 \times 2\) dimensions
• Matrix \((B+C)\):\(2 \times 2\) dimensions

since A has 2 columns and \((B+C)\) has 2 rows, you can multiply them.
The resulting matrix will have the dimensions of the outer numbers, i.e., \(3 \times 2\). For instance:

• First row, first column: The sum of \(4 \times 6 + 0 \times 0 = 24\)
• First row, second column: The sum of \(4 \times 0 + 0 \times (-1) = 0\)

Repeat for each row and column to get the final result: \(A(B+C) = \begin{bmatrix} 24 & 0 \ -18 & -5 \ 0 & -1 \end{bmatrix}\).
Understanding this multiplication process is crucial for deeper topics in linear algebra.

Dot Product

The dot product is a fundamental concept used in matrix multiplication. It's performed for each element in the resulting matrix by taking one row from the first matrix and one column from the second and multiplying corresponding elements, then summing them up.
Consider Matrix A and Matrix \((B+C)\) here:

• Dot product for the first row, first column in the resulting matrix: \((4 \cdot 6) + (0 \cdot 0) = 24\)
• Dot product for the second row, second column: \((-3 \cdot 0) + (5 \cdot -1) = -5\)

To perform this, ensure you obey the inner dimension rule—where the number of columns in the first matrix equals the number of rows in the second matrix.
Learning the dot product is not just useful in matrix multiplication but also plays a critical role in various applications like physics and computer science. It helps in calculating projections, understanding vector orientations, and is foundational in more advanced operations such as matrix inversions and determinants.