Problem 25 - College Algebra with Corequisite Support

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. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(4 A+5 D\)

Step-by-Step Solution

Answer

The operation is not possible because matrices A and D have different dimensions.

Step 1: Matrix Dimensions

First, determine the dimensions of matrices A and D. Matrix A is a 2x2 matrix, and matrix D is a 3x3 matrix.

Step 2: Understand Scalar Multiplication

Matrices can be multiplied by numbers (scalars) that are outside the matrix. For example, 4A would mean each element of A should be multiplied by 4.

Step 3: Verify Addition of Matrices

To add matrices, both must have the same dimensions. Matrix A is 2x2, and matrix D is 3x3. Since their dimensions are unequal, we cannot perform the addition.

Step 4: Conclusion

The operation "4A + 5D" is not possible because matrices A and D have different dimensions and cannot be added together.

Key Concepts

Matrix Addition

Matrix addition is an operation where you combine two matrices to create a new matrix. To perform matrix addition, both matrices must have the same dimensions. This means they must have the same number of rows and columns. For example, if you have a 2x2 matrix, you can only add it to another 2x2 matrix.
When adding matrices, you add their corresponding elements. For example, if you have matrices \(A\) and \(B\):
\[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]
Their sum \(C\) would be:
\[C = A + B = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}\]
As the solution indicated, matrix \(A\) is 2x2 and matrix \(D\) is 3x3, so they cannot be added. An operation like \(4A + 5D\) is impossible due to these differing dimensions, highlighting the importance of understanding matrix addition rules before attempting such tasks.

Matrix Dimensions

Matrix dimensions are crucial to understanding operations involving matrices. The dimension of a matrix is described as its number of rows by its number of columns. It is usually represented in the format: rows x columns.
For instance, a matrix with two rows and two columns is a 2x2 matrix.

• Matrix \(A\) in the exercise is a 2x2 matrix
• Matrix \(D\) is a 3x3 matrix

When working with matrices, aligning dimensions correctly is fundamental. For addition and subtraction, matrices must have "identical" dimensions. However, this isn't a requirement for matrix multiplication, where the number of columns in the first matrix must match the number of rows in the second.
By understanding and identifying matrix dimensions correctly, you can avoid mistakes in various operations such as addition or multiplication. The exercise example with \(4A + 5D\) clearly showcases the importance of recognizing that mismatched dimensions block addition.

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a single number, known as a scalar. This operation scales the entire matrix by the scalar's value.
Suppose you have a matrix \(A\): \[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\]And you wish to multiply it by a scalar \(k\), the result is: \[kA = \begin{bmatrix} k \cdot a_{11} & k \cdot a_{12} \ k \cdot a_{21} & k \cdot a_{22} \end{bmatrix}\]Scalar multiplication is straightforward because it simply involves basic arithmetic with each matrix element.
In the exercise, one operation was to multiply matrix \(A\) by 4 (\(4A\)) and matrix \(D\) by 5 (\(5D\)). Each respective element of these matrices would be scaled by their scalar. However, despite executing these multiplications correctly, the results can't be summed up due to incompatible dimensions for matrix addition, as was highlighted in the step-by-step solution.