Question

Solve for \(X\) when $$A=\left[\begin{array}{rr} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{array}\right].$$ $$ 2 X+3 A=B $$

Short answer

The matrix \(X\) is found by first performing a scalar multiplication of \(A\) by 3, then subtracting this from matrix \(B\), and finally dividing the result by 2.

Step-by-step solution

Apply the scalar multiplication

First, calculate the result of \(3A\). Each element of the matrix \(A\) is multiplied by 3.

Subtract the calculated scalar multiple of \(A\) from \(B\)

Next, subtract \(3A\) from \(B\). The result is \(2X\).

Division by scalar to isolate \(X\)

To isolate \(X\), the results from step 2 should be divided by 2 which gives us the final result for \(X\).

Key concepts

Matrix Operations

Matrix operations include several fundamental procedures, such as addition, subtraction, and multiplication, which allow for the manipulation and transformation of matrices. These operations adhere to specific rules that must be followed to achieve correct results.
One important element of matrix operations is the need for matrices to have the same dimensions to perform addition or subtraction. This means that you can only add or subtract two matrices if they have the same number of rows and columns. If the dimensions don't match, these operations simply cannot be performed.

• Addition and Subtraction: You add or subtract matrices element-wise. This involves taking each corresponding element from the two matrices and performing the operation.
• Multiplication: Multiplying matrices is more complex than addition and subtraction. When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second.

Additionally, the scalar multiplication of a matrix involves multiplying every element by a constant, which we'll explore more in the next section. Understanding these operations is essential for solving matrix equations.

Scalar Multiplication

Scalar multiplication in matrix algebra involves multiplying every element of a matrix by a scalar, which is a single number. This operation is straightforward but requires careful arithmetic to ensure accuracy.
In the exercise, scalar multiplication is used to calculate \(3A\). To do this, each element of matrix \(A\) is multiplied by 3. For example, if \(A\) has an element \(a_{ij}\), then in \(3A\), that element becomes \(3 \times a_{ij}\).
This operation affects the entire matrix uniformly and is a key component in the manipulation of matrices.

• This technique helps in scaling the matrix, similar to stretching or shrinking an image while keeping its proportions.
• Scalar multiplication is one of the foundational operations that must be understood to work with more complex matrix equations.

Once \(3A\) is computed, it sets the stage for further operations like subtraction in solving matrix equations.

Matrix Equation

A matrix equation typically involves finding an unknown matrix that satisfies the equation, much like solving for \(x\) in algebra. In the given problem, the matrix equation is \(2X + 3A = B\).
This type of equation requires applying matrix operations systematically to isolate the unknown matrix \(X\).

• Steps to solve:
• First, calculate \(3A\) by performing scalar multiplication on matrix \(A\).
• Subtract \(3A\) from \(B\) to simplify the equation to \(2X = B - 3A\).
• Finally, divide every element of the resulting matrix by 2 to find \(X\), as it's the result of dividing the matrix \(2X\) by the scalar 2.

This sequence of operations highlights the importance of understanding both scalar multiplication and matrix operations. By systematically applying these steps, the solution to the matrix equation reveals the unknown matrix, offering insights into how different matrices interact within the constraints of the equation.