Problem 23 - College Algebra

Question

Let $$A=\left[\begin{array}{rr}-3 & -7 \\\2 & -9 \\\5 & 0\end{array}\right] \text { and } B=\left[\begin{array}{rr}-5 & -1 \\\0 & 0 \\\3 & -4\end{array}\right]$$ Solve each matrix equation for \(X\). $$B-X=4 A$$

Step-by-Step Solution

Answer

The matrix \(X\) that satisfies the equation \(B -X = 4A\) is \(X = \left[\begin{array}{rr}7 & 27 \\ -8 & 36 \\ -17 & -4\end{array}\right]\).

Step 1: Write the given equation

To start, write down the equation provided in the exercise: \(B - X = 4A\).

Step 2: Rewrite the equation for \(X\)

Rearrange the equation with the intention of isolating \(X\) on one side. This can be done by adding \(X\) to both sides and subtracting \(4A\) from both sides to get: \(X = B - 4A\).

Step 3: Substitute the known matrices into the equation

Substitute the given matrices \(A\) and \(B\) into the equation. \(X = B - 4A = \left[ \begin{array}{rr}-5 & -1 \\0 & 0 \\3 & -4\end{array}\right] - 4\left[\begin{array}{rr}-3 & -7 \\2 & -9 \\5 & 0\end{array}\right]\).

Step 4: Calculate 4A

Calculate \(4A\) by multiplying every element in matrix \(A\) by 4: \(4A = \left[\begin{array}{rr}-12 & -28 \\8 & -36 \\20 & 0\end{array}\right]\).

Step 5: Subtract \(4A\) from \(B\)

Subtract matrix \(4A\) from matrix \(B\) by subtracting the corresponding elements: \(X = B - 4A = \left[\begin{array}{rr}-5 -(-12) & -1 -(-28) \\0 -8 & 0 -(-36) \\3 -20 & -4 -0\end{array}\right]\).

Step 6: Calculate the elements in \(X\)

Perform the subtraction operation for each element to obtain \(X\): \(X = \left[\begin{array}{rr}7 & 27 \\ -8 & 36 \\ -17 & -4\end{array}\right]\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying rows of the first matrix by columns of the second matrix. The product of two matrices, say matrix \(A\) and matrix \(B\), is possible only if the number of columns in \(A\) is equal to the number of rows in \(B\). This operation results in a new matrix whose dimensions are the number of rows in \(A\) by the number of columns in \(B\). To perform matrix multiplication:

• Take each row of the first matrix.
• Multiply its elements with the corresponding elements of a column in the second matrix.
• Add the products to get the final value which is placed in the resultant matrix.

When dealing with large matrices, careful alignment or use of computational software can help avoid mistakes. Matrix multiplication is not commutative, meaning \(AB \, eq \, BA\) in most cases. Understanding this concept is crucial, especially when solving matrix equations and transformations.

Matrix Subtraction

Matrix subtraction is an operation similar to the concept of subtracting numbers. However, it applies to matrices by subtracting corresponding elements. For two matrices to be subtracted, they must have the same dimensions. Here's how matrix subtraction works:

• The matrices should have the same number of rows and columns.
• Subtract the corresponding elements from each entry in the matrices.
• The resulting matrix will have the same dimensions as the original matrices.

In the provided exercise, after calculating \(4A\) by scaling matrix \(A\), matrix \(B\) is subtracted from this scaling to isolate matrix \(X\). It's important to subtract each element accurately, observing the signs to avoid errors. Just like addition, subtraction of matrices is straightforward once this principle is adhered to.

Linear Algebra

Linear algebra is an essential branch of mathematics that focuses on linear equations, matrices, determinants, vectors, and vector spaces. Matrix operations, as seen in the exercise, are core components of linear algebra. They allow us to handle systems of linear equations efficiently.
Its applications span various fields such as engineering, physics, computer science, and statistics. In linear algebra:

• Matrices represent systems of linear equations.
• Operations like matrix multiplication and subtraction help solve these equations.
• Concepts like eigenvalues, eigenvectors, and transformations are fundamental.

Understanding linear algebra enables one to solve complex equations and perform data manipulation, critical for real-world applications like signal processing and machine learning. Mastery of matrix operations and understanding of vectors, subspaces, and transformations give a strong foundation in this discipline.