Problem 17 - 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 \(X\). $$ X-A=B $$

Step-by-Step Solution

Answer

The matrix \(X\) is thus calculated as: \[ X = \begin{bmatrix} -8 & -8 \ 2 & -9 \ 8 & -4 \end{bmatrix}\]

Step 1: Understand the Matrix equation

The equation to solve is \(X - A = B\). Here, \(X\), \(A\) and \(B\) are matrices. The solution is to isolate \(X\) on one side. This means we need to add matrix \(A\) to both sides of the equation.

Step 2: Add Matrix \(A\) to Both Sides of equation

Matrix \(A\) is added to both sides of the equation to isolate matrix \(X\). The new equation will be \(X = B + A\)

Step 3: Calculate \(B + A\)

To add two matrices together, add corresponding elements together. Therefore, add \(-3\) to \(-5\), \(-7\) to \(-1\), \(2\) to \(0\), \(-9\) to \(0\), \(5\) to \(3\), and \(0\) to \(-4\). The sum will be the matrix \(X\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, employed when combining two matrices of the same dimension. Understanding how to perform matrix addition is key to solving many problems involving matrices. Imagine having two arrays of numbers, organized in rows and columns – these are matrices. When we add them, we simply line up the numbers in matching positions and add them together, one pair at a time.

For instance, given two matrices:
\begin{align*}A &= \begin{bmatrix}a_{11} & a_{12} \ a_{21} & a_{22}\rend{bmatrix}, \ B &= \begin{bmatrix}b_{11} & b_{12} \b_{21} & b_{22}\rend{bmatrix},d{align*}the sum, denoted as \( C = A + B \), would be:
\begin{align*}C &= \begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12} \a_{21} + b_{21} & a_{22} + b_{22}\rend{bmatrix}.d{align*}In real practice, just like the example from the textbook exercise, you align both matrices and add the numbers that occupy the same 'spot' in each matrix to get the result. Always ensure the matrices are of the same dimension before trying to add them, or else the operation is undefined.

Algebraic Operations on Matrices

Just like in algebra with numbers, we can perform various operations on matrices including addition, subtraction, multiplication, and scalar multiplication. These operations are bound by rules that maintain the structure of a matrix.

• Addition: We've already seen how matrix addition works. It's a matter of adding corresponding entries.
• Subtraction: This is similar to addition, but instead, you subtract the corresponding entries.
• Scalar Multiplication: Instead of adding or subtracting another matrix, here you multiply each entry in the matrix by a numerical value (a scalar).
• Matrix Multiplication: This is a more complex operation where we calculate the dot product of rows and columns across two matrices. It's not as straightforward as addition or subtraction, and it can only be done when the number of columns in the first matrix is equal to the number of rows in the second matrix.

Each of these operations has specific rules that must be obeyed, without which the operation may not be valid. For the exercise in question, we focused on adding matrices to both sides of an equation to isolate a variable – which brings us to our next topic.

Isolating Variables in Matrices

The process of isolating a variable matrix is akin to what we do in basic algebra. When given an equation where the variable is not by itself, we perform operations that 'cancel out' other elements in order to get that variable alone on one side of the equation. In the context of matrices, this often involves using matrix addition or subtraction.When isolating the matrix \(X\) in the equation \(X - A = B\), we want to 'move' \(A\) over to the other side with \(B\). In algebraic terms, we effectively add \(A\) to both sides:\begin{align*}X - A + A &= B + A \X &= B + Ad{align*}It's crucial to note that the order of addition does not matter in matrices as they follow the commutative property of addition (\(A + B = B + A\)). Isolating variables is a powerful technique that allows us to solve matrix equations systematically and understand better the underlying relationships between different matrices in a given system.