Problem 24 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Matrix Addition and Subtraction

For matrix addition or subtraction to be possible, the matrices involved must have the same dimensions. Matrices A, B, and C are all 2x2 matrices, so we can perform addition and subtraction on these matrices.

Step 2: Add Matrices A and B

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

Step 3: Subtract Matrix C from Result

Now subtract matrix C from the result (\(A + B\)), ensuring to subtract corresponding elements. Compute \((A + B) - C\):\[(A + B) - C = \begin{bmatrix} -7 & 1 \ 2 & 9 \end{bmatrix} - \begin{bmatrix} 0 & 9 \ 7 & 1 \end{bmatrix} = \begin{bmatrix} -7 - 0 & 1 - 9 \ 2 - 7 & 9 - 1 \end{bmatrix} = \begin{bmatrix} -7 & -8 \ -5 & 8 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices by adding their corresponding elements. To perform matrix addition, it's essential that both matrices have the same dimensions.
For instance, if you're adding two 2x2 matrices:

• Matrix A: \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• Matrix B: \( \begin{bmatrix} e & f \ g & h \end{bmatrix} \)

You simply add corresponding elements to get the resulting matrix:\[A + B = \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix}\]In the case where matrices don't have the same dimensions, addition isn't possible. This is because each element in the first matrix needs a corresponding element in the second matrix for the operation to be well-defined.
Matrix addition is commutative, meaning that the order does not matter, i.e., \(A + B = B + A\). This property simplifies many computational processes.

Matrix Subtraction

Matrix subtraction is very similar to matrix addition, but instead of adding the elements, you subtract them. As with addition, the matrices involved must have the same dimensions.
Consider the subtraction of matrix C from matrix A, where:

• Matrix A: \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• Matrix C: \( \begin{bmatrix} e & f \ g & h \end{bmatrix} \)

The resulting matrix, after subtracting each corresponding element, would be:\[A - C = \begin{bmatrix} a-e & b-f \ c-g & d-h \end{bmatrix}\]This operation is not commutative. Unlike addition, changing the order will result in a different matrix, i.e., \(A - C eq C - A\).
Matrix subtraction is useful in many applications, especially when determining the difference between two data sets represented as matrices.

Matrix Dimensions

Matrix dimensions refer to the number of rows and columns that a matrix has. Understanding dimensions is crucial as it dictates possible operations and compatibility between matrices.
The dimension of a matrix is represented as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. For example:

• A matrix with 2 rows and 3 columns is a \(2 \times 3\) matrix.

In the exercise, matrices A, B, and C are all \(2 \times 2\) matrices, allowing for both matrix addition and subtraction between them. Conversely, attempting to add or subtract a \(2 \times 2\) matrix with a \(2 \times 3\) matrix (like matrix D) results in an error since the dimensions do not match.
Being conscious of matrix dimensions ensures smooth computational operations and prevents errors in matrix algebra.