Problem 11 - Algebra 2

Question

Solve each matrix equation. $$ \left[\begin{array}{rrr}{2} & {1} & {-1} \\ {0} & {2} & {1}\end{array}\right]-X=\left[\begin{array}{rrr}{11} & {3} & {-13} \\ {15} & {-9} & {8}\end{array}\right] $$

Step-by-Step Solution

Answer

\left[\begin{array}{rrr}{2} & {1} & {-1} \ {0} & {2} &{1}\end{array}\right] - X = \left[\begin{array}{rrr}{11} & {3} & {-13} \ {15} &{-9} & {8}\end{array}\right] \Rightarrow X = \left[\begin{array}{rrr}{-9} & {-2} & {12} \ {-15} & {11} & {-7}\end{array}\right]

Step 1: Identify the Matrix Equation

First, recognize that the matrix equation is in the form of \(A - X = B\), where \(A\) is the given matrix on the left, \(X\) is the unknown matrix, and \(B\) is the given matrix on the right.

Step 2: Isolate the Unknown Matrix

To solve for \(X\), add matrix \(A\) to both sides of the matrix equation to isolate \(X\) on one side: \(X = A - B\).

Step 3: Calculate the Subtraction of Matrices

Subtract matrix \(B\) from matrix \(A\) element-wise to find matrix \(X\). Remember that the subtraction of matrices involves subtracting corresponding elements from each other.

Step 4: Write the Resulting Matrix

Perform the subtraction for each element of the matrices to obtain the resulting matrix \(X\).

Key Concepts

Matrix Subtraction

Understanding matrix subtraction is crucial when working with matrix equations. This operation is performed element-wise, meaning you subtract each element of one matrix from the corresponding element of another matrix. For instance, if you have two matrices, \( A \) and \( B \), both of the same dimensions, their subtraction \( A - B \) results in a new matrix where each element \( a_{ij} - b_{ij} \) is the subtraction of elements located at the same position \( (i,j) \) in matrices \( A \) and \( B \).

To visualize, consider the matrix:\[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23}\end{array}\] and subtracting another matrix of the same size:\[\begin{array}{ccc}b_{11} & b_{12} & b_{13} \b_{21} & b_{22} & b_{23}\end{array}\], the result would be:\[\begin{array}{ccc}a_{11} - b_{11} & a_{12} - b_{12} & a_{13} - b_{13} \a_{21} - b_{21} & a_{22} - b_{22} & a_{23} - b_{23}\end{array}\].
It's essential to remember that you can only subtract matrices of the same dimensions; otherwise, the operation is undefined.

Isolate the Unknown Matrix

To solve matrix equations, it often becomes necessary to isolate the unknown matrix on one side of the equation. In algebraic terms, this is similar to solving for 'x' in basic algebra, except instead of an unknown variable, we have an unknown matrix. For example, if we have \( A - X = B \), where \( A \) and \( B \) are known matrices, and \( X \) is the unknown, we would rearrange the equation to solve for \( X \) by adding \( A \) to both sides. The resulting isolated unknown matrix will then be \( X = A - B \), setting the stage for the actual computation using matrix subtraction.

This step is where the algebra of matrices mirrors the algebra of numbers, keeping in mind that matrix addition and subtraction follow specific rules that must be adhered to, such as the requirement for matrices to be of the same dimension.

Element-Wise Matrix Operations

Element-wise matrix operations are a foundational aspect of matrix algebra. These operations imply that you perform the same arithmetic operation on each corresponding pair of elements from the two matrices involved. For subtraction and addition, this means calculating the result for each element position individually.

For example, if we subtract one matrix from another, we subtract the elements in the first row and first column of each matrix, then move to the first row and second column, and so on until all positions have been processed. It's important to note that for element-wise operations, the matrices must match in size; you can't perform these operations on matrices of differing dimensions. Element-wise operations make up the core of simpler matrix manipulations and are used extensively in various mathematical and practical applications.

Algebraic Methods in Matrices

When dealing with matrices, algebraic methods are employed to solve equations, just as with single-variable algebraic equations. These methods include addition, subtraction, multiplication, and division (the latter through multiplication by an inverse matrix, if it exists). Such methods are used to rearrange matrix equations, isolate unknowns, and ultimately solve for specific matrices or values.

In the context of our initial problem, algebraic methods are applied to both isolate the unknown matrix \( X \) and perform the necessary computations to solve the equation. Understanding these methods is key to manipulating and solving more complex matrix equations which can involve multiple unknowns and a greater number of matrix operations. Familiarity with these concepts can open doors to advanced topics in linear algebra such as vector spaces, determinants, and eigenvalues.