Problem 26 - College Algebra

Question

In Exercises \(17-26,\) 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 \(X\). $$ 4 B+3 A=-2 X $$

Step-by-Step Solution

Answer

The matrix X is found by doing scalar multiplication of 4 with matrix B, 3 with matrix A, adding these two resulting matrices together, and then dividing this sum by \(-2\).

Step 1: Compute the Result of Scalar Multiplication and Addition

First, we need to calculate 4B and 3A and then add these two matrices together. Scalar multiplication is performed by multiplying each element in the matrix by the scalar value. Matrix addition is performed by adding each corresponding item in the two matrices.

Step 2: Evaluate -2X

Next, we need to isolate \(X\) by dividing both sides of the equation by \(-2\). Just like in scalar multiplication, division involves dividing each element in the matrix by the scalar value.

Step 3: Resulting Matrix X

The resulting matrix after executing the operations is the matrix \(X\). This is the solution to the matrix equation.

Key Concepts

Scalar Multiplication

Scalar multiplication is a basic matrix operation that involves multiplying each element within a matrix by a scalar value. In our original exercise, this process was performed twice: once as 4 times matrix B, and then as 3 times matrix A. Here’s how it's done in a simple, straightforward way. To perform scalar multiplication:

• Select a scalar value. In this case, they are '4' and '3'.
• Multiply each element of the chosen matrix by the scalar value. For example, multiplying each component of matrix A by 3 means multiplying -3, -7, 2, -9, 5, and 0 by 3 accordingly.

The operation transforms each element from the original matrix in proportion to the scalar, effectively "scaling" the matrix by that factor. It's essential to carry out precise calculations, as any mistake in scalar multiplication can affect the final result of the matrix equation.

Matrix Addition

Matrix addition is another fundamental operation, crucial in solving matrix equations. It involves adding corresponding elements of matrices to attain a resultant matrix. In the exercise, after computing 4B and 3A using scalar multiplication, these results are added together. Key steps in matrix addition include:

• Ensure both matrices are of the same size, which means they must have the same number of rows and columns.
• Add each pair of corresponding elements from the matrices.

For example, adding the transformed matrices (from the scalar multiplication step) involves simply summing each pair of elements. If the element of 4B was 'x' and the element of 3A was 'y', their sum will be 'x + y'. This operation is typically straightforward but demands attention to detail to ensure accuracy across all entries of the matrices.

Matrix Division

In the context of matrices, division is less intuitive than scalar multiplication or matrix addition. You can't directly divide matrices like you divide numbers, but you can divide a matrix by a scalar by multiplying by the reciprocal of the scalar. In our problem, after obtaining the summed matrix from adding 4B and 3A, the process to solve for X was to divide the whole matrix by -2: Here's how to think about matrix division:

• Identify the matrix you want to "divide", in our case, the matrix sum.
• Consider dividing all elements by the scalar. This effectively means multiplying them by the reciprocal of the scalar.

For the scalar -2, you multiply each element of the matrix by -1/2. This division algorithmically resembles multiplying each entry by (1/-2), streamlining the process while preserving the integrity of the matrix's structure during equation resolution.