Problem 25 - College Algebra

Question

Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{rr}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(4 A+5 D\)

Step-by-Step Solution

Answer

The operation \(4A + 5D\) is not possible due to different dimensions of matrices \(A\) and \(D\).

Step 1: Understand the Requirements

The problem is asking us to perform the matrix operation \(4A + 5D\). This involves scalar multiplication and matrix addition.

Step 2: Confirm Matrix Dimensions

Matrix \(A\) is a \(2 \times 2\) matrix and matrix \(D\) is a \(3 \times 3\) matrix. To add two matrices, they must have the same dimensions.

Step 3: Determine Operation Feasibility

Since matrix \(A\) is \(2 \times 2\) and matrix \(D\) is \(3 \times 3\), the addition of these two matrices is not possible because their dimensions do not match.

Step 4: Conclusion

The operation \(4A + 5D\) cannot be performed due to mismatched dimensions of matrices \(A\) and \(D\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a basic operation in matrix algebra. It involves multiplying each element of a matrix by a single number, known as a scalar. If you have a matrix and you want to scale its values, scalar multiplication is the way to go.

• To perform scalar multiplication, simply take the scalar and multiply it by every entry in the matrix.
• If a matrix is denoted by \(A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\), and the scalar is \(k\), then \(kA\) is \(\begin{bmatrix} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{bmatrix}\).

This operation does not change the size of the matrix; it only scales the elements, keeping the number of rows and columns the same.
In our original exercise, the step of multiplying matrices \(A\) and \(D\) by 4 and 5 respectively is an example of scalar multiplication.

Matrix Addition

Matrix addition is another fundamental operation where we combine two matrices by adding their corresponding elements. For this operation to be possible, the matrices must be the same size.
Here's how it works:

• If you have two matrices of the same dimensions, \(A\) and \(B\), the sum \(A + B\) is the matrix formed by adding corresponding elements: \(a_{ij} + b_{ij}\).
• For example, if \(A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \), then \(A + B = \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}\).
• Matrix addition is only defined for matrices of the same dimensions.

In our exercise, since matrices \(A\) and \(D\) do not have the same dimensions, adding \(4A and 5D\) is not feasible.

Matrix Dimensions

Matrix dimensions refer to the size of a matrix, which is determined by the number of rows and columns it contains. A matrix with \(m\) rows and \(n\) columns is called an \(m \times n\) matrix.
Knowing the dimensions of a matrix is crucial for performing various operations:

• The dimensions determine whether two matrices can be added or multiplied together.
• For example, a \(2 \times 3\) matrix looks like \(\begin{bmatrix} a & b & c \ d & e & f \end{bmatrix}\).

In the context of the original exercise, matrix \(A\) is a \(2 \times 2\) matrix, and matrix \(D\) is a \(3 \times 3\) matrix.
The mismatch in dimensions means these matrices cannot be added, as they do not have the same shape.

Addition Feasibility

The feasibility of adding matrices depends on their dimensions. For matrix addition to be possible, both matrices must have identical dimensions, meaning the same number of rows and columns.
Whenever you face a matrix addition problem, always:

• Check the dimensions of both matrices first.
• Ensure they match. If one is \(2 \times 2\) and the other is \(3 \times 3\), like matrices \(A\) and \(D\) in our exercise, addition cannot be performed.
• If they match, add the corresponding elements.

Thus, checking addition feasibility is a vital step in operations involving matrices. The inability to add \(4A + 5D\) in the problem arose precisely because the matrices had different dimensions.