Problem 24 - 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. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {6} & {7}\end{array}\right], B=\left[\begin{array}{cc}{-9} & {6} \\ {-4} & {2}\end{array}\right], C=\left[\begin{array}{cc}{0} & {9} \\ {7} & {1}\end{array}\right], D=\left[\begin{array}{rrr}{-8} & {7} & {-5} \\ {4} & {3} & {2} \\ {0} & {9} & {2}\end{array}\right], E=\left[\begin{array}{rrr}{4} & {5} & {3} \\ {7} & {-6} & {-5} \\ {1} & {0} & {9}\end{array}\right] $$ $$ A+B-C $$

Step-by-Step Solution

Answer

The result is \(\begin{bmatrix} -7 & -8 \\ -5 & 8 \end{bmatrix}\).

Step 1: Verify if Matrix Addition is Possible

Matrices can only be added if they have the same dimensions. Matrix \(A\) is \(2\times2\), matrix \(B\) is \(2\times2\), and matrix \(C\) is also \(2\times2\). As all matrices are \(2\times2\), addition is possible.

Step 2: Add Matrices A and B

To add matrix \(A\) and matrix \(B\), add the corresponding elements:\[A + B =\begin{bmatrix} 2 + (-9) & -5 + 6 \ 6 + (-4) & 7 + 2 \end{bmatrix} = \begin{bmatrix} -7 & 1 \ 2 & 9 \end{bmatrix}\]

Step 3: Subtract Matrix C from the Result

Subtract the corresponding elements of matrix \(C\) from the result obtained in Step 2:\[(A + B) - C =\begin{bmatrix} -7 - 0 & 1 - 9 \ 2 - 7 & 9 - 1 \end{bmatrix} = \begin{bmatrix} -7 & -8 \ -5 & 8 \end{bmatrix}\]

Step 4: Conclusion

The result of the operation \(A + B - C\) is \(\begin{bmatrix} -7 & -8 \ -5 & 8 \end{bmatrix}\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It involves summing two or more matrices by adding their corresponding elements. This operation is straightforward and only requires matrices to have the same dimensions. For example, if matrix A is a 2x2 matrix, matrix B must also be a 2x2 matrix for them to be added together. If they do not match in dimension, addition is not possible.
Matrix addition operates element-wise: the item in row i, column j of the resulting matrix is the sum of the items in row i, column j of the original matrices.
For instance, if we are given:

• Matrix A: \[\begin{bmatrix} 2 & -5 \ 6 & 7 \end{bmatrix}\]
• Matrix B: \[\begin{bmatrix} -9 & 6 \ -4 & 2 \end{bmatrix}\]

We compute each corresponding pair:

• Top left: 2 + (-9) = -7
• Top right: -5 + 6 = 1
• Bottom left: 6 + (-4) = 2
• Bottom right: 7 + 2 = 9

The sum of matrices A and B will then be:\[\begin{bmatrix} -7 & 1 \ 2 & 9 \end{bmatrix}\]This result highlights how we simply add each corresponding element from the two matrices.

Matrix Subtraction

Matrix subtraction is similar in operation to matrix addition but involves subtracting one matrix from another. Just like addition, subtraction requires matrices to match in dimension. When you subtract matrices, you subtract each corresponding element from the first matrix by the corresponding element in the second matrix.
Let's say we want to subtract Matrix C from the result of adding Matrix A and Matrix B. Given the matrix from addition:\[\begin{bmatrix} -7 & 1 \ 2 & 9 \end{bmatrix}\]and Matrix C:\[\begin{bmatrix} 0 & 9 \ 7 & 1 \end{bmatrix}\],we perform the operation by:

• Top left: \(-7 - 0 = -7\)
• Top right: \(1 - 9 = -8\)
• Bottom left: \(2 - 7 = -5\)
• Bottom right: \(9 - 1 = 8\)

Resulting in:\[\begin{bmatrix} -7 & -8 \ -5 & 8 \end{bmatrix}\]This subtraction process reaffirms how each element is individually handled.

Matrix Dimensions

Matrix dimensions are a key factor in several matrix operations including addition, subtraction, and multiplication. These dimensions describe the size of a matrix in terms of its rows and columns.
For instance, a matrix with 2 rows and 3 columns is labeled as a 2x3 matrix. The order is important: rows are listed first, columns second.
In our example:

• Matrix A is a 2x2 matrix.
• Matrix B is a 2x2 matrix.
• Matrix C is also a 2x2 matrix.
• Matrix D has a different shape, being a 3x3 matrix.

For operations such as addition and subtraction, all matrices involved must be of the same dimensions. If not, operations will be undefined due to mismatched sizes. Checking and confirming matrix dimensions is a crucial first step before performing any operations. It ensures that your calculations are possible and that you're applying the operations correctly.