Problem 12 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

No solution exists because \(C\) and \(B\) have different dimensions.

Step 1: Understand the Equation

We are given the matrix equation \(3X - B = C\). Our objective is to solve for the unknown matrix \(X\). We will isolate \(X\) by adding \(B\) to both sides, resulting in \(3X = C + B\). After that, we will solve for \(X\) by dividing the resultant matrix by 3.

Step 2: Add Matrices B and C

Calculate the sum of matrices \(C\) and \(B\). First, check if their dimensions allow for addition: \(C\) is a \(3 \times 2\) matrix and \(B\) is a \(2 \times 2\) matrix. Since they don't have the same dimension, addition is not possible, which implies the equation has no solution.

Key Concepts

Understanding Matrix Addition

Matrix addition is a fundamental operation in linear algebra and involves adding corresponding elements from two matrices. For the addition of two matrices to be valid, the matrices must have the same dimensions. This means that both matrices need to have the same number of rows and columns.
For example, two matrices, both with dimensions of 2x2, can be added together by adding their corresponding entries. If a matrix has an element in the first row and first column of the value 4, and another matrix has a corresponding element of value 3, the resulting matrix after addition will have an element with the value 7 in the same position.

• Matrix A: \(\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\)
• Matrix B: \(\begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\)

The result of adding A and B would then be: \(\begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}\)
Remember, if the dimensions do not match, like in the matrices from our exercise (C is 3x2 and B is 2x2), matrix addition is not possible.

Matrix Dimensions Explained

When working with matrices, understanding dimensions is crucial. A matrix's dimension is written as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. These dimensions are important because they determine if certain operations, like addition and multiplication, can be performed.
For effective matrix addition, both matrices must have the same number of rows and columns. For example, if one matrix is \(2 \times 3\), it must be added to another \(2 \times 3\) matrix. Breaking this down, if a matrix has 2 rows and 3 columns (\(2 \times 3\)), it means:

• There are 2 rows
• There are 3 columns

Understanding this concept is critical when examining the matrices B and C from the problem. B is \(2 \times 2\) and C is \(3 \times 2\). Since their dimensions differ, they cannot be directly added, highlighting one important reason why the matrix equation \(3X - B = C\) has no solution, as the operations implied require consistent dimensions.

Solving Matrix Equations

Matrix equations are like regular algebraic equations but involve matrices. The key steps to solving a matrix equation like \(3X - B = C\) include isolating the unknown matrix. Start by rearranging the equation to solve for \(X\). We add matrix B to both sides to get \(3X = C + B\).
When trying to perform such operations, always check matrix dimensions. This is to ensure the operations are valid. As we learned from earlier sections, our attempt to add matrices B and C failed due to incompatible dimensions, signaling no solution exists.
If dimensions were suitable and we obtained \(3X = D\) where D is some resultant matrix after addition, the next step involves solving for \(X\) by multiplying both sides by \(\frac{1}{3}\) (effectively dividing the matrix D by 3, element-wise), which gives \(X = \frac{1}{3} D\). When performing such operations, keep in mind each element of the matrix is scaled by multiplying by \(\frac{1}{3}\).
Understanding these operations helps in effectively tackling matrix equations and spotting when no solution exists due to dimensional mismatches.