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

Step-by-Step Solution

Answer

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

Step 1: Identify the Operation

We need to compute the expression \(A + B - C\). This involves performing both matrix addition and subtraction on the matrices \(A\), \(B\), and \(C\).

Step 2: Check Matrix Dimensions

Ensure that all matrices involved in the operation \(A + B - C\) have the same dimensions. Matrix \(A\) is \(2 \times 2\), \(B\) is \(2 \times 2\), and \(C\) is \(2 \times 2\). All matrices have the same dimensions, so the operations are possible.

Step 3: Perform Addition of Matrices

Add matrices \(A\) and \(B\):\[A + B = \begin{bmatrix} 2 & -5 \ 6 & 7 \end{bmatrix} + \begin{bmatrix} -9 & 6 \ -4 & 2 \end{bmatrix} = \begin{bmatrix} 2 + (-9) & -5 + 6 \ 6 + (-4) & 7 + 2 \end{bmatrix} = \begin{bmatrix} -7 & 1 \ 2 & 9 \end{bmatrix}\]

Step 4: Perform Subtraction of Matrix

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

Step 5: Final Result

The final result of the operation \(A + B - C\) is the matrix:\[\begin{bmatrix} -7 & -8 \ -5 & 8 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is an essential operation in linear algebra that combines two matrices by adding their corresponding elements. To add two matrices, they must have the same dimensions, meaning the same number of rows and columns.
For example, if you have matrices \(A\) and \(B\) both of size \(2 \times 2\), you can add them directly. Here's how the addition is performed:

• Take matrix \(A = \begin{bmatrix} 2 & -5 \ 6 & 7 \end{bmatrix}\) and matrix \(B = \begin{bmatrix} -9 & 6 \ -4 & 2 \end{bmatrix}\).
• Add each element from matrix \(A\) to the corresponding element in matrix \(B\).
• The result is \(A + B = \begin{bmatrix} (2) + (-9) & (-5) + 6 \ 6 + (-4) & 7 + 2 \end{bmatrix} = \begin{bmatrix} -7 & 1 \ 2 & 9 \end{bmatrix}\).

After the addition, you get a new matrix where each element is simply the sum of elements from the original matrices at that position. Clear understanding of this step is crucial before proceeding to matrix subtraction.

Matrix Subtraction

Matrix subtraction is very similar to matrix addition but involves subtracting each corresponding pair of elements from two matrices. Just like with addition, the matrices must be of the same dimensions to carry out this operation.
Let's continue with our example, starting from the resulting matrix from an addition as \(A + B = \begin{bmatrix} -7 & 1 \ 2 & 9 \end{bmatrix}\), and matrix \(C = \begin{bmatrix} 0 & 9 \ 7 & 1 \end{bmatrix}\).

• Subtract each element of matrix \(C\) from the corresponding element in the added result \((A + B)\).
• This gives \((A + B) - C = \begin{bmatrix} -7 - 0 & 1 - 9 \ 2 - 7 & 9 - 1 \end{bmatrix} = \begin{bmatrix} -7 & -8 \ -5 & 8 \end{bmatrix}\).

With matrix subtraction, you are finding the difference between the elements at each position in the matrices. This operation can be thought of as adding a matrix of negative values.

Matrix Dimensions

Matrix dimensions are fundamental to understanding whether two matrices can be added or subtracted. They refer to the number of rows and columns in a matrix. For instance, a \(2 \times 2\) matrix has two rows and two columns, while a \(3 \times 3\) matrix has three rows and three columns.
Checking dimensions is a prerequisite for addition and subtraction of matrices because:

• Only matrices of the exact same dimensions can be added or subtracted. Incompatible dimensions result in operations being impossible.
• If matrix \(A\) is \(m \times n\), any matrix added to or subtracted from \(A\) must also be \(m \times n\).

In our exercise with matrices \(A\), \(B\), and \(C\), they all have dimensions \(2 \times 2\). This consistency allows us to proceed with the addition and subtraction freely. Remember always to verify matrix dimensions first when preparing to perform these operations.