Question

Matrices \(A\) and \(B\) are given below. Find \(X\) that satisfies the equation. $$ A=\left[\begin{array}{cc} 3 & -1 \\ 2 & 5 \end{array}\right] \quad B=\left[\begin{array}{cc} 1 & 7 \\ 3 & -4 \end{array}\right] $$ $$ A-X=3 B $$

Short answer

X = \( \begin{bmatrix} 0 & -22 \\ -7 & 17 \end{bmatrix} \) .

Step-by-step solution

Understand the Equation

The given equation is \( A - X = 3B \). We need to solve this equation for the matrix \( X \). This involves substituting the given matrices \( A \) and \( B \) into the equation and then isolating \( X \).

Calculate 3B

To compute \( 3B \), multiply every element of matrix \( B \) by 3:\[3B = 3 \begin{bmatrix} 1 & 7 \ 3 & -4 \end{bmatrix} = \begin{bmatrix} 3 \times 1 & 3 \times 7 \ 3 \times 3 & 3 \times (-4) \end{bmatrix} = \begin{bmatrix} 3 & 21 \ 9 & -12 \end{bmatrix}\]

Substitute 3B into the Equation

Substitute the matrix \( 3B \) into the equation \( A - X = 3B \):\[\begin{bmatrix} 3 & -1 \ 2 & 5 \end{bmatrix} - X = \begin{bmatrix} 3 & 21 \ 9 & -12 \end{bmatrix}\]

Isolate the Matrix X

To isolate \( X \), rearrange the equation to \( X = A - 3B \). This can be done by adding \( -3B \) to both sides of the equation. Therefore, we have:\[X = \begin{bmatrix} 3 & -1 \ 2 & 5 \end{bmatrix} - \begin{bmatrix} 3 & 21 \ 9 & -12 \end{bmatrix}\]

Perform the Subtraction

Subtract corresponding elements of the two matrices to determine \( X \):\[X = \begin{bmatrix} 3-3 & -1-21 \ 2-9 & 5-(-12) \end{bmatrix} = \begin{bmatrix} 0 & -22 \ -7 & 17 \end{bmatrix}\]

Final Step: State the Result

The matrix \( X \) that satisfies the equation \( A - X = 3B \) is:\[X = \begin{bmatrix} 0 & -22 \ -7 & 17 \end{bmatrix}\]

Key concepts

Matrix Multiplication

Matrix multiplication is fundamental in solving matrix equations. In simple terms, it involves multiplying two matrices by finding the dot product of the rows and columns. To understand matrix multiplication:

• Each element of the resultant matrix is computed by taking the sum of products of corresponding elements of a row of the first matrix and a column of the second matrix.
• The number of columns in the first matrix must equal the number of rows in the second matrix.
• This results in a new matrix with dimensions derived from the rows of the first and columns of the second matrix.

When multiplying a matrix by a scalar, such as in our exercise with calculating \( 3B \):

• Multiply every element within the matrix by the scalar value (in this case, 3).
• This process scales the matrix, giving each element a new value determined by the scalar multiplication.

Matrix Subtraction

Matrix subtraction involves subtracting one matrix from another, element by element. To subtract matrices:

• Ensure the matrices involved have the same dimensions. This is crucial because matrix subtraction, like addition, is not defined for matrices of unequal sizes.
• Subtract each element of the second matrix from the corresponding element of the first matrix.

In our example, we needed to calculate the subtraction \( A - 3B \):

• Compare matrices that have the same structure allowing each corresponding element to be subtracted.
• For example, the entry in the first row and column of \( X \) is found by subtracting the entry in the same position in \( 3B \) from the entry in \( A \).

This process results in a new matrix that reflects the difference between the two matrices on an element-wise basis.

Solving Matrix Equations

Solving matrix equations often requires manipulating and rearranging the equation to isolate the unknown matrix, much like solving algebraic equations. Key steps to solve a matrix equation:

• Identify the components of the equation, like known matrices and the matrix to be solved (e.g., \( X \)).
• Substitute the values of known matrices into the equation to simplify.
• Use operations like addition, subtraction, scalar multiplication, and sometimes matrix multiplication to isolate the unknown matrix.

In the original exercise, the equation \( A - X = 3B \) required isolation of \( X \) by rearranging to \( X = A - 3B \).
This involves performing matrix subtraction once the form \( X = A - (3B) \) was determined, thus enabling an easy path to find \( X \).
Always check the dimensions and compatibility of operations being performed to ensure the solution is valid and consistent with matrix rules.