Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving a Matrix Equation Solve for \(X\) when \(A=\left[\begin{array}{rr}-2 & -1 \\\ 1 & 0 \\ 3 & -4\end{array}\right]\) and \(B=\left[\begin{array}{rr}0 & 3 \\\ 2 & 0 \\ -4 & -1\end{array}\right]\) $$-3 X-3 A=9 B$$

Step-by-Step Solution

Answer

The matrix \(X\) will be obtained by performing the operations of scalar multiplication and addition on the given matrices \(A\) and \(B\). The exact value of \(X\) will depend on the given values of \(A\) and \(B\).

Step 1: Analyze the initial equation

The original equation is -3X - 3A = 9B. Here, the goal is to solve for \(X\). So one should start by isolating terms involving \(X\).

Step 2: Isolate the terms with the X matrix

Add 3A to both sides of the equation. This gives: -3X = 9B + 3A.

Step 3: Divide both sides by -3

To solve for matrix \(X\), both sides of the equation should be divided by -3. This gives the final equation: X = -(9B + 3A) / 3.

Step 4: Simplify the equation

When simplifying this equation, it leads to X = -3B - A as each entry in matrices \(A\), \(B\) and \(X\) is treated separately. Follow the rules of matrix addition and scalar multiplication to compute this.

Step 5: Calculate the X matrix

Replacing \(A\) and \(B\) with their respective given values and performing the operations, the matrix \(X\) is obtained.

Key Concepts

Matrix Scalar Multiplication

Understanding matrix scalar multiplication is essential when dealing with equations involving matrices. This operation involves multiplying every entry of a matrix by the same scalar, which is a constant number. For instance, if we have a matrix C and we multiply it by the scalar 3, written as 3C, we multiply every element within the matrix C by 3.

This operation is straightforward yet critical, as it affects each component of the matrix identically. In the provided exercise, scalar multiplication is used when dividing both sides of the equation by -3, to isolate the matrix X. Since division by a scalar is equivalent to multiplying by its reciprocal, we multiply every element in the matrices 9B and 3A by -1/3 in this step.

Matrix Addition

Matrix addition is another fundamental operation used in the manipulation of matrix equations. The rule is simple: to add two matrices, they must be of the same size; then, you add the corresponding elements together. This method is applied element-wise, meaning you take each element from the first matrix and add it to the corresponding element in the second matrix.

For example, assume matrices D and E have the same dimensions. The sum of D + E will yield a new matrix where each entry is the sum of entries D_ij + E_ij for all i, j. It's crucial in our exercise after isolating matrix X, -3X = 9B + 3A requires adding matrices B and A together after scalar multiplication.

Isolating Variables in Matrices

Isolating the variable in a matrix equation is the process of manipulating the equation to get the unknown matrix, often denoted as X, by itself on one side of the equation. This can be done using a series of operations like matrix addition or scalar multiplication to systematically simplify and rearrange the terms.

In the given problem, to isolate X, we start by adding 3A to both sides of the equation -3X - 3A = 9B, which eliminates the A term from one side. The next step is to divide by the scalar -3, which involves scalar multiplication by the reciprocal. By performing these operations, which adhere to the same basic arithmetic principles as with regular numbers, the variable X is isolated and can then be computed by substituting the values given for A and B.