Problem 14 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Set Up the Equation

The problem states the equation is \(5(X-C) = D\). Our goal is to isolate \(X\). To start, let's clearly write the equation and identify the matrices involved: \(X\), \(C\), and \(D\).

Step 2: Solve for \(X-C\)

To isolate \(X-C\) on one side of the equation, divide both sides of the equation by 5. This gives us: \[ X - C = \frac{1}{5}D \] Readablity wise it's equivalent to have each element of matrix \(D\) multiplied by \(1/5\).

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

Calculate \(\frac{1}{5}D\), which means multiplying each element of matrix \(D\) by \(\frac{1}{5}\). So, \(\frac{1}{5}D\) becomes: \[\frac{1}{5} \times \begin{bmatrix} 10 & 20 \ 30 & 20 \ 10 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 4 \ 2 & 0 \end{bmatrix}\]

Step 4: Solve for \(X\)

Add matrix \(C\) to both sides of the equation \(X - C = \begin{bmatrix} 2 & 4 \ 6 & 4 \ 2 & 0 \end{bmatrix}\). \(C\) is \(\begin{bmatrix} 2 & 3 \ 1 & 0 \ 0 & 2 \end{bmatrix}\). So, \(X = \begin{bmatrix} 2 & 4 \ 6 & 4 \ 2 & 0 \end{bmatrix} + \begin{bmatrix} 2 & 3 \ 1 & 0 \ 0 & 2 \end{bmatrix}\).

Step 5: Calculate \(X\)

Add the corresponding elements of the two matrices from the previous step. Thus, \(X\) becomes: \[X = \begin{bmatrix} 2+2 & 4+3 \ 6+1 & 4+0 \ 2+0 & 0+2 \end{bmatrix} = \begin{bmatrix} 4 & 7 \ 7 & 4 \ 2 & 2 \end{bmatrix}\].

Key Concepts

Matrix Multiplication

Matrix multiplication is a way to combine two matrices to create a new matrix. In matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This means not all pairs of matrices can be multiplied together. For example, if matrix \( A \) is \( m \times n \), then matrix \( B \) must be \( n \times p \) for the multiplication \( AB \) to be defined. The resulting matrix will be \( m \times p \).

An important step in matrix operations is to ensure that matrices are dimensionally compatible before attempting operations like multiplication. Unlike regular multiplication, matrix multiplication is not commutative, meaning \( AB eq BA \). Each element in the resulting matrix is computed by taking the sum of products of the corresponding elements from the rows of the first matrix and the columns of the second matrix. To better understand the construction, visualize each element as

• Row element of the first matrix times column element of the second matrix.
• Sum these products to obtain the element in the new matrix.

Matrix multiplication also plays a critical role in transformations and systems of equations, making it essential to master.

Matrix Addition

Matrix addition is one of the simpler matrix operations, and it involves adding two matrices element-wise. In order to add two matrices, they must be of the same dimensions, meaning they both have the same number of rows and columns. This ensures that there is a corresponding element in each matrix to add together.

The process of matrix addition is straightforward:

• Add the elements in the same position from each matrix.
• Place the resulting sum in the corresponding position in the new matrix.

For example, if matrices \( A \) and \( B \) are both \( 2 \times 2 \) matrices, then the sum \( A + B \) will also be a \( 2 \times 2 \) matrix.

Matrix addition is often used in conjunction with other matrix operations, such as in solving matrix equations like the one in the original exercise, where matrix \(C\) is added to the simplified form of the equation to solve for \(X\). Understanding how to perform accurate matrix addition is fundamental, especially when working towards determining unknown matrices.

Matrix Division

Matrix division is not defined in the same straightforward manner as scalar division. Instead, it generally involves multiplying by the inverse of a matrix. For a matrix \( A \), its inverse \( A^{-1} \) exists only if \( A \) is a square matrix and its determinant is not zero. The operation to 'divide' by a matrix is performed by multiplying by its inverse.

For a matrix equation like \( AX = B \), you would solve for \( X \) by multiplying both sides by \( A^{-1} \):

• \( X = A^{-1}B \).

However, not all matrices have inverses; this is determined by calculating the determinant. If it is zero, the matrix is singular and non-invertible, meaning you cannot perform the division-like operation.

In our original problem, we indirectly addressed matrix division by isolating \( X \) with scalar division when we divided each element of matrix \( D \) by 5. This simplifies handling the divisions that occur in matrix operations.

Solving Matrices

Solving matrices often involves using several matrix operations to isolate an unknown matrix in an equation. This process may include steps like matrix division, addition, and multiplication. When faced with an equation such as \( 5(X-C) = D \), the objective is to find the matrix \( X \) that satisfies the equation.

The approach taken in solving for \( X \) begins with understanding the order of operations required. Follow these steps:

• Isolate the expression containing the unknown (\( X-C \)) by undoing scalar multiplication if needed.
• Utilize matrix division concepts by dividing or distributing the scalar across the matrix. In this case, we divided \( D \) by 5.
• Employ matrix addition to finally solve for \( X \) by adding the matrix \( C \) to both sides of the equation.

This systematic approach ensures each operation contributes distinctly towards unraveling the unknown matrix. Such practices underscore the importance of undertaking these operations in the correct sequence to maintain the integrity of the mathematical relationships involved.