Problem 15 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The solution for the unknown matrix \(X\) is \(\begin{bmatrix}0 & -5 \\ -25 & -20 \\ -10 & 10 \end{bmatrix}.\)

Step 1: Multiply Equation by 5

To solve for \(X\), we first eliminate the fraction from the equation \(\frac{1}{5}(X+D) = C\) by multiplying both sides by 5, which gives us the equation: \(X + D = 5C\).

Step 2: Multiply Matrix C by 5

Calculate \(5C\) by multiplying every entry of matrix \(C\) by 5. The resulting matrix is \[5C = \begin{bmatrix} 2 \cdot 5 & 3 \cdot 5 \ 1 \cdot 5 & 0 \cdot 5 \ 0 \cdot 5 & 2 \cdot 5 \end{bmatrix} = \begin{bmatrix} 10 & 15 \ 5 & 0 \ 0 & 10 \end{bmatrix}.\]

Step 3: Subtract D from 5C

To isolate \(X\), subtract matrix \(D\) from \(5C\). This gives us \(X = 5C - D\). Substitute the matrices from previous steps: \[X = \begin{bmatrix} 10 & 15 \ 5 & 0 \ 0 & 10 \end{bmatrix} - \begin{bmatrix} 10 & 20 \ 30 & 20 \ 10 & 0 \end{bmatrix}.\]

Step 4: Compute the Subtraction

Subtract corresponding elements of \(D\) from \(5C\) to find \(X\): \[X = \begin{bmatrix} 10 - 10 & 15 - 20 \ 5 - 30 & 0 - 20 \ 0 - 10 & 10 - 0 \end{bmatrix} = \begin{bmatrix} 0 & -5 \ -25 & -20 \ -10 & 10 \end{bmatrix}.\]

Step 5: Confirm the Dimensions Match

Verify that the dimensions of the resultant matrices in all operations are compatible; recall that both \(5C\) and \(D\) are \(3 \times 2\) matrices, so the subtraction is valid and produces a \(3 \times 2\) matrix for \(X\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a potent operation that allows us to combine matrices in a specific way. Unlike regular arithmetic, multiplying matrices requires attention to the size or dimensions of the matrices involved.
Matrix multiplication is possible only when the number of columns in the first matrix equals the number of rows in the second matrix. For instance, if you have a matrix with dimensions 2x3, it can be multiplied by a matrix with dimensions 3x4.
Notably, multiplying a matrix by a scalar (like multiplying Matrix C by 5 in the exercise) means multiplying each element of the matrix by that scalar.
When handling matrix multiplication:

• Ensure the dimensions are compatible.
• Calculate each element in the resulting matrix by summing the products of rows and columns.
• Understand that matrix multiplication is not commutative, meaning A * B is not necessarily the same as B * A.

Matrix Subtraction

Matrix subtraction is quite straightforward in comparison to multiplication. It involves direct subtraction of corresponding elements from two matrices.
For matrix subtraction to be valid, both matrices must have identical dimensions. This means every element in one matrix must directly correspond to an element in the other matrix.
In the exercise, we found matrix X by subtracting matrix D from 5C. This subtraction involved:

• Ensuring both 5C and D are 3x2 matrices, making their subtraction feasible.
• Subtracting each corresponding element from D out of 5C to get the resulting matrix.

Remember:
- Matrix subtraction, like addition, is commutative when applied element-wise (A - B equals -(B - A)).
- Always check dimensions are equal before performing operations.

Matrix Dimensions

In any matrix operations, knowing the dimensions of the matrices involved is crucial. Matrix dimensions are described using two numbers: rows x columns.
Let's break down why this is vital.

• Dimensions dictate possible operations; e.g., the arrays resulting from possible matrix multiplications, or ensuring addition/subtraction are viable.
• In matrix equations, like in the step-by-step solution of the exercise, examining dimensions ensures compatibility between matrices undergoing operations, such as ensuring 5C and D both have dimensions 3x2.

Also, dimensions are significant because they define the shape of the resulting matrix after operations.
Whether adding, subtracting, or multiplying matrices, always perform dimension checks to prevent errors. Recognizing dimensions early helps anticipate the form of the solution.