Problem 30 - College Algebra

Question

Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(A B\)

Step-by-Step Solution

Answer

Matrix multiplication is possible; the resulting matrix is \( \begin{bmatrix} -800 & 500 \\ -300 & 800 \end{bmatrix} \).

Step 1: Determine the dimensions of matrices A and B

Matrix A is a 2x2 matrix, and matrix B is a 2x2 matrix. Therefore, both matrices have dimensions of 2 rows and 2 columns.

Step 2: Confirm compatibility for multiplication

Since the number of columns in matrix A (2 columns) is equal to the number of rows in matrix B (2 rows), the matrices A and B can be multiplied.

Step 3: Perform matrix multiplication

To multiply matrices A and B, use the formula for each element \((i, j)\) in the resulting matrix: \((AB)_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj}\).For the first row, first column element: \((AB)_{11} = (-10) \times 40 + 20 \times (-20) = -400 - 400 = -800\).For the first row, second column element: \((AB)_{12} = (-10) \times 10 + 20 \times 30 = -100 + 600 = 500\).For the second row, first column element: \((AB)_{21} = 5 \times 40 + 25 \times (-20) = 200 - 500 = -300\).For the second row, second column element: \((AB)_{22} = 5 \times 10 + 25 \times 30 = 50 + 750 = 800\).

Step 4: Write the resulting matrix

The resulting matrix from the multiplication of matrices A and B is:\[ AB = \begin{bmatrix} -800 & 500 \ -300 & 800 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

When working with matrices, understanding their dimensions is key. The dimensions of a matrix are defined by the number of rows and columns it has, typically expressed as "rows x columns." For example, a matrix with 2 rows and 3 columns is a "2x3 matrix". Knowing these dimensions helps us understand how the matrix looks, and more importantly, if certain operations like multiplication are feasible.
Matrix A, as per the exercise, is a 2x2 matrix which means it has 2 rows and 2 columns. This indicates a simple structure where every element can be easily paired with another matrix of matching column and row dimensions.

Matrix Compatibility

Matrix multiplication isn't always possible, and checking if matrices are compatible for multiplication is a crucial initial step. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. This rule ensures connectedness, as each entry in one matrix needs a corresponding entry to multiply in another.
In our example, matrix A has 2 columns, and matrix B has 2 rows, meaning these matrices are compatible. It’s essential to always validate this, because even a perfect-looking setup could face issues if dimensions don’t align. When dimensions do match, as here, you can proceed confidently with the operation.

Matrix Operations

Matrix multiplication involves some intricate operations but can be broken down into simple repeated steps. First, consider each element in the resulting matrix individually. Each element is calculated by taking a row from the first matrix and a column from the second. Multiply these corresponding elements and add them up.
Use the formula \( (AB)_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj} \) for calculating these elements, where \( i \) represents the row index and \( j \) the column index in the resulting matrix. This method is repetitive across the matrix, ensuring each element in the new matrix is accurately created.
Practice this by walking through the matrix row by row and column by column.

Resulting Matrix

The outcome of matrix multiplication is another matrix, often with a different dimension than the original matrices, reflecting the crossed rows and columns. In our exercise, the operation \( AB \) results in a 2x2 matrix, which matches the row count of matrix A and the column count of matrix B, confirming the successful multiplication as expected.
The resulting matrix is given by \[ AB = \begin{bmatrix} -800 & 500 \ -300 & 800 \end{bmatrix} \]. Here, each element has been systematically produced by applying the multiplication rule across every row-column pair. It's important to review the process for each element to ensure all operations were executed correctly.
This reinforced understanding can help in verifying results and reinforcing conceptual clarity.