Problem 24 - College Algebra

Question

Let $$A=\left[\begin{array}{rr}-3 & -7 \\\2 & -9 \\\5 & 0\end{array}\right] \text { and } B=\left[\begin{array}{rr}-5 & -1 \\\0 & 0 \\\3 & -4\end{array}\right]$$ Solve each matrix equation for \(X\). $$A-X=4 B$$

Step-by-Step Solution

Answer

The matrix \(X\) is found by performing arithmetic operations on matrices \(A\) and \(B\). After rearranging the equation, we subtract matrix \(4B\) from matrix \(A\) to solve for matrix \(X\).

Step 1: Multiply Matrix B by 4

First, multiply every element in matrix B by 4. Let's refer to this new matrix as \(4B\).

Step 2: Rearrange the Equation

We can rearrange the equation to be \(X = A - 4B\), which means that to find \(X\), we need to subtract the modified matrix \(4B\) from \(A\).

Step 3: Perform Matrix Subtraction

Matrix subtraction is performed elementwise. This means we subtract the corresponding element in \(4B\) from the one in \(A\) to form the corresponding element in \(X\).

Key Concepts

Matrix Multiplication

Matrix multiplication involves the scaling and adding of matrix rows and columns. It requires strict compliance to specific rules. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second.

• Matrix A (of size m x n) can be multiplied by matrix B (of size n x p) to produce a new matrix of size m x p.
• This operation requires multiplying elements from the rows of the first matrix by corresponding elements in the columns of the second matrix and then summing the results.

In the given exercise, multiplying matrix B by 4 is a simpler form of matrix multiplication. Here, each individual element in matrix B is multiplied by the scalar 4:

• Multiply each element in matrix B, \( \left[ \begin{array}{rr} -5 & -1 \ 0 & 0 \ 3 & -4 \end{array} \right] \), by 4.
• This operation results in a new matrix: \( 4B = \left[ \begin{array}{rr} -20 & -4 \ 0 & 0 \ 12 & -16 \end{array} \right] \).

Despite the simplicity here with scalar multiplication, understanding how to manage rows and columns is essential for more complex matrix operations.

Matrix Subtraction

Matrix subtraction operates on elementwise principles, similar to matrix addition. For two matrices to be subtracted, they must share identical dimensions. This ensures each value at a specific location in one matrix directly aligns with its counterpart in the other.

• For matrices A and B, with dimensions m x n, matrix subtraction produces a result matrix of size m x n.
• Each element in the resulting matrix is computed by subtracting the element from matrix B from its corresponding element in matrix A.

Using the given exercise, we perform matrix subtraction after calculating 4B:

• Given matrices: \( A = \left[ \begin{array}{rr} -3 & -7 \ 2 & -9 \ 5 & 0 \end{array} \right] \) and \( 4B = \left[ \begin{array}{rr} -20 & -4 \ 0 & 0 \ 12 & -16 \end{array} \right] \).
• Subtract each corresponding element: \( A - 4B = \left[ \begin{array}{rr} -3 - (-20) & -7 - (-4) \ 2 - 0 & -9 - 0 \ 5 - 12 & 0 - (-16) \end{array} \right] \).
• This eventually leads to a result of \( X = \left[ \begin{array}{rr} 17 & -3 \ 2 & -9 \ -7 & 16 \end{array} \right] \).

Thorough understanding of these interactions between matrices builds a foundation for more advanced matrix equations.

Elementwise Operations

Elementwise operations are fundamental in matrix algebra, providing a straightforward way to handle matrices. For any matrix operation to be considered elementwise, each matrix must be of the same dimension, allowing calculations to line up row by row, column by column.

• The term "elementwise" implies that operations occur between individual elements located at the same position in two matrices.
• This is applicable in both matrix addition and subtraction where direct element correspondence is crucial.

In our problem, using elementwise subtraction for matrices A and 4B means performing the following for each pairwise matching element:

• Corresponding values are subtracted: \( a_{ij} - b_{ij} \) for each element position \((i, j)\).
• This results in a new matrix reflecting the individual differences across all matching elements.

Understanding elementwise operations ensures a solid grasp of how different matrix arithmetic combines to solve matrix equations effectively.