Problem 13 - Algebra 2

Question

Solve each matrix equation. $$ X+\left[\begin{array}{rr}{6} & {1} \\ {-2} & {3}\end{array}\right]=\left[\begin{array}{rr}{2} & {0} \\ {-3} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

\[\begin{array}{rr}{-4} & {-1} \ {-1} & {-2}\end{array}\]

Step 1: Identify the matrices and the equation

Firstly, identify the given matrix equation and the matrices involved. Let's call the unknown matrix X, and the given matrices A and B, where A = \[\begin{array}{rr}{6} & {1} \ {-2} & {3}\end{array}\] and B = \[\begin{array}{rr}{2} & {0} \ {-3} & {1}\end{array}\]. The matrix equation can be written as X + A = B.

Step 2: Solve for the matrix X

To find matrix X, we need to subtract matrix A from both sides of the equation, thus isolating X. The subtraction of two matrices is performed by subtracting the corresponding entries. This works out to be X = B - A.

Step 3: Perform the matrix subtraction

Perform the element-wise subtraction B - A to find the entries of matrix X. After this calculation, X should be equal to \[\begin{array}{rr}{2 - 6} & {0 - 1} \ {-3 - (-2)} & {1 - 3}\end{array}\], which simplifies to \[\begin{array}{rr}{-4} & {-1} \ {-1} & {-2}\end{array}\].

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental operation in linear algebra that involves the element-wise difference of two matrices of the same dimensions. To perform matrix subtraction, simply subtract the corresponding elements of one matrix from the other. For instance, if we have two matrices, C and D, the subtraction C - D is done by subtracting each element of D from the corresponding element in C.

In the provided exercise, we're given two 2x2 matrices, A and B, and instructed to solve for the unknown matrix X in the equation X + A = B. Subtracting matrix A from both sides, we align each element of A under its corresponding element of B and perform the subtraction. This process, which is similar to elementary arithmetic, provides us with the unknown matrix X.

Matrix Operations

Matrix operations, including addition, subtraction, multiplication, and division (in the form of inverting matrices), serve as a cornerstone to many applications in mathematics, engineering, and science. These operations must follow specific rules. For example, addition and subtraction require matrices of the same size, and positional elements are combined directly.

However, when it comes to multiplication, the number of columns in the first matrix must be equal to the number of rows in the second. Unlike simple arithmetic, matrix operations are not commutative in general; this means that A + B is not always equal to B + A, especially for matrix multiplication. Understanding these rules is crucial for correctly solving matrix equations and for the broader application of matrices in solving systems of equations or transforming geometrical figures.

Algebraic Matrices

Algebraic matrices are matrices that contain algebraic numbers (numbers that can be expressed as polynomials) or variable entries that can be used to represent systems of linear equations. In the case of the exercise provided, we're dealing with constant numbers inside the matrices, but the principles are the same when algebraic expressions are involved.

The solutions to algebraic matrix equations are found by employing matrix operations. In this context, when an equation involves an unknown matrix, solving it entails isolating the unknown, much like you would with regular algebraic equations involving numbers. Once isolated, the resulting matrix expression can be simplified using the appropriate matrix operations to reveal the unknown matrix. This practice is fundamental in myriad mathematical fields and applications and it's an invaluable skill for students studying linear algebra.