Problem 18 - Algebra and Trigonometry

Question

Solve the matrix equation for the unknown matrix \(X,\) or explain why no solution exists. $$ \begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right] \quad D=\left[\begin{array}{cc}{10} & {20} \\\ {30} & {20} \\ {10} & {0}\end{array}\right]}\end{array} $$ $$ 5(X-C)=D $$

Step-by-Step Solution

Answer

The solution is the matrix \(X = \begin{bmatrix} 4 & 7 \\ 7 & 4 \\ 2 & 2 \end{bmatrix}\).

Step 1: Understand the Matrix Equation

The given matrix equation is \(5(X - C) = D\). Our goal is to find the unknown matrix \(X\). We will first isolate \(X\) in the equation.

Step 2: Rearrange the Equation

To find \(X\), we need to isolate it. Start by dividing both sides of the equation by 5 to get \(X-C = \frac{1}{5}D\).

Step 3: Calculate \(\frac{1}{5}D\)

Compute \(\frac{1}{5}D\) by multiplying each element of matrix \(D\) by \(\frac{1}{5}\): \[ \frac{1}{5}D = \begin{bmatrix} \frac{10}{5} & \frac{20}{5} \ \frac{30}{5} & \frac{20}{5} \ \frac{10}{5} & \frac{0}{5} \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 4 \ 2 & 0 \end{bmatrix} \]

Step 4: Solve for X

Now that we have \(X-C = \begin{bmatrix} 2 & 4 \ 6 & 4 \ 2 & 0 \end{bmatrix}\), add matrix \(C\) to both sides to solve for \(X\). Since \(C = \begin{bmatrix} 2 & 3 \ 1 & 0 \ 0 & 2 \end{bmatrix}\), calculate: \[ X = \begin{bmatrix} 2 & 4 \ 6 & 4 \ 2 & 0 \end{bmatrix} + \begin{bmatrix} 2 & 3 \ 1 & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix} 4 & 7 \ 7 & 4 \ 2 & 2 \end{bmatrix} \]

Step 5: Verify the Dimensions of the Matrices

Finally, check that the calculated matrix \(X\) has appropriate dimensions. Here, matrices must be compatible in addition and subtraction. The resulting matrix \(X\), \(\begin{bmatrix} 4 & 7 \ 7 & 4 \ 2 & 2 \end{bmatrix}\), is 3x2 as required, confirming the solution is correct.

Key Concepts

Solving Matrix Equations

Solving matrix equations can initially seem daunting, but breaking down the process into stages simplifies the task. In this scenario, the matrix equation is given by \(5(X-C)=D\). To find the solution for the unknown matrix \(X\), follow these steps:

• First, it’s vital to isolate \(X\). Begin by dividing both sides by 5 to remove the scalar multiplying the entire equation: \(X - C = \frac{1}{5}D\).
• The next move is to solve the simpler equation for \(X\). This entails adding matrix \(C\) to both sides, leading to \(X = \frac{1}{5}D + C\).

By carrying out these operations with care, expressing \(X\) in terms of known matrices and operations becomes straightforward.
Remember, each arithmetic operation involving matrices adheres to specific rules so precision in performing each step is essential for the correct solution.

Matrix Addition and Subtraction

Understanding matrix addition and subtraction is crucial to solving matrix equations efficiently. Here's a quick guide:

• Matrix addition involves adding corresponding elements from two matrices of the same dimension. For example, if you have matrices \(A\) and \(B\), each element \(a_{ij}\) from \(A\) is added to \(b_{ij}\) from \(B\).
• Simultaneously, matrix subtraction works similarly. The element \(c_{ij}\) from matrix \(C\) is subtracted from the corresponding element \(d_{ij}\) in \(D\).

The solution to the equation from Step 4 involves adding matrix \(C\) to the matrix resulting from \(\frac{1}{5}D\). It's important that both matrices share the same dimensions to be combined, allowing element-wise addition. In our problem, once both operations are performed correctly, the resulting matrix \(X\) is achieved.
Practicing these operations on smaller matrices can help generate familiarity, easing the path to mastering their application in more complex scenarios.

Matrix Dimensions

The dimensions of matrices are a fundamental characteristic and greatly influence the outcome of any operation involving them. Matrices are typically denoted by their row and column count, \(m \times n\), where \(m\) stands for the number of rows and \(n\) for columns.

• It's crucial that matrices being added or subtracted share identical dimensions. Failing this, operations such as addition or subtraction simply can't be conducted.
• For multiplication, the number of columns in the first matrix must equal the number of rows in the second. If not, the product will be undefined.

In our particular problem, maintaining awareness of dimensions at each step ensures operations are valid and meaningful. We examined matrix \(C\), \(\frac{1}{5}D\), and their sum to make sure they are compatible at each stage. This vigilance confirms the calculated matrix \(X\) remains consistent with required dimensions.
With practice, awareness of these dimensions becomes second nature, and helps in both planning operations and verifying results for accuracy.