Problem 25 - 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 \(X\). $$ 4 A+3 B=-2 X $$

Step-by-Step Solution

Answer

The matrix \(X\) is obtained by the operations described in the steps above and it represents the solution of the given equation.

Step 1: Write out matrices

First, write out the given matrices \(A\) and \(B\) and the equation including these matrices, \(4A + 3B = -2X\).

Step 2: Perform the multiplications and addition on the left side

Next, multiply every number in matrix \(A\) with 4 and every number in matrix \(B\) with 3 then add up the corresponding elements from these two results. This gives us a new matrix that represents the left side of the equation.

Step 3: Solve for the matrix \(X\)

As we want to isolate \(X\), we need to divide both sides of the equation by -2. We divide every element in the resulting matrix from the last step with -2. This results in the matrix \(X\).

Key Concepts

Algebraic Matrices

Algebraic matrices are essentially grids of numbers or expressions arranged in rows and columns, which can represent data or mathematical relationships. Think of them as numerical arrays where each element has a specific position, defined by its row and column. In algebra, these matrices can be added, subtracted, and multiplied, and can also be used to solve systems of linear equations.

When dealing with algebraic matrices, it's important to remember that the size of the matrix matters. The size is determined by the number of rows and columns it contains, often denoted as 'm x n' where 'm' stands for rows and 'n' for columns. For our exercise, both matrices A and B are of size '3 x 2', meaning they each have three rows and two columns. In many operations involving two matrices, like addition or subtraction, they must be of the same size for the operation to be possible.

Matrix Operations

Matrix operations include various ways in which we can manipulate these grids of numbers. The most common operations include addition, subtraction, multiplication, and finding the inverse or determinant of a matrix.

When we multiply a matrix by a scalar, as seen in our exercise, we simply multiply every element in the matrix by that number. It's like scaling up or down all the numbers in our matrix grid. For instance, multiplying matrix A by 4 means we'll multiply each number in A by 4. Similarly, multiplying matrix B by 3 means the same for B.

Scalar Multiplication

Scalar multiplication alters all elements of the matrix uniformly, and it's a crucial step before adding or subtracting matrices in some problems, including our example.

Matrix Addition

Matrix addition is adding two matrices by adding their corresponding elements. It's like combining two grids of numbers into one, position by position. For instance, if we have two matrices C and D, the sum would be a new matrix E where each element of E is the sum of the corresponding elements from C and D.

To add matrices, they must be of the same size; otherwise, we can't match up the elements to add them. It's akin to combining two puzzles; if they're not the same size, the pieces just won't fit together. Let's connect it back to our task: after multiplying matrices A and B by their respective scalars, we perform matrix addition by adding the resulting matrices element-wise to get the left side of our equation.

Example of Matrix Addition

If we have element A₁₁ in matrix A and the corresponding element B₁₁ in matrix B, their sum will be the element of the resulting matrix C in the same position, i.e., C₁₁ = A₁₁ + B₁₁. This addition process is carried out for every corresponding element in matrices A and B.