Problem 17 - Algebra and Trigonometry

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. $$ X-A=B $$

Step-by-Step Solution

Answer

The solution to the matrix equation is \(X = \left[\begin{array}{rr}{-8} & {-8} \ {2} & {-9} \ {8} & {-4}\end{array}\right]\)

Step 1: Identify The Matrices

The matrices in the equation are \(A = \left[\begin{array}{rr}{-3} & {-7} \ {2} & {-9} \ {5} & {0}\end{array}\right]\) and \(B = \left[\begin{array}{rr}{-5} & {-1} \ {0} & {0} \ {3} & {-4}\end{array}\right]\)

Step 2: Rearrange The Equation

We want to isolate \(X\). To do this, we need to remove \(A\) from the left of the equation, this results in \(X = A + B\).

Step 3: Add The Matrices Together

Now, we add matrix \(A\) and matrix \(B\) together, using element-by-element addition. Recall that the element in the \(i\)th row and \(j\)th column of the resultant matrix, is the sum of the elements in the \(i\)th row and \(j\)th column in both \(A\) and \(B\). So, \(X = \left[\begin{array}{rr}{-3} & {-7} \ {2} & {-9} \ {5} & {0}\end{array}\right] + \left[\begin{array}{rr}{-5} & {-1} \ {0} & {0} \ {3} & {-4}\end{array}\right] = \left[\begin{array}{rr}{-3 + (-5)} & {-7 + (-1)} \ {2 + 0} & {-9 + 0} \ {5 + 3} & {0 + (-4)}\end{array}\right]\)

Key Concepts

Matrix Operations

Matrix operations are fundamental concepts in linear algebra that involve the manipulation of arrays of numbers. Matrices can be added, subtracted, or multiplied, and each operation follows specific rules. Matrix addition, subtraction, and scalar multiplication are straightforward as they operate element-wise, meaning that each number in a matrix interacts with the corresponding number in another matrix or scalar. For matrix multiplication, when two matrices are multiplied together, the resulting matrix is produced by taking the dot product of rows and columns.

• **Addition and Subtraction**: Possible only if matrices are of the same size.
• **Multiplication**: Involves more complex rules related to the dimensions of matrices involved.
• **Scalar Multiplication**: Each element of the matrix is multiplied by a scalar (a single number).

Understanding these operations is essential, as they form the basis of manipulating data in algebraic equations.

Element-Wise Addition

Element-wise addition is one of the simplest matrix operations. It involves adding corresponding elements of two matrices. For element-wise operations to take place, both matrices must be of the same dimensions, meaning they have the same number of rows and columns.

In our exercise, matrix A and matrix B are both 3x2 matrices and are compatible for addition.

• **Step 1**: Start by aligning the matrices.
• **Step 2**: Add the elements from the top left corner, going to the right, and then down each row.
• **Step 3**: Ensure the resulting matrix retains the same dimensions, maintaining a 3x2 size just like A and B.

This operation is crucial in many applications, such as solving linear equations and performing transformations in graphics.

Algebraic Structures

In mathematics, an algebraic structure refers to a set equipped with one or more operations that satisfy a list of axioms. Matrices can form algebraic structures. For example, when matrices are combined with addition and scalar multiplication, they create a structure similar to a vector space.

Studying algebraic structures involving matrices helps us understand how these objects behave under various operations. Some important structures are:

• **Vector Spaces**: A set of vectors, which regard matrices as objects that can be added together and multiplied by scalars.
• **Groups**: Regarding certain aspects of matrix multiplication and inverses.

By exploring these algebraic structures, learners can appreciate matrices' expressive power and relevance in broader mathematical contexts.

Matrices in Algebra

Matrices play a pivotal role in algebra due to their ability to represent systems of linear equations, transformations, and perform computations efficiently. They provide a compact way to handle big sets of numbers and facilitate operations that would be cumbersome to perform element by element without them.

• **Representation of Systems**: Matrices encapsulate equations in a concise format, allowing easy manipulation.
• **Transformations**: In many applications, matrices are used to perform linear transformations, which are crucial in computations dealing with space and geometry.

As a fundamental algebraic tool, matrices help solve equations, represent data transformations, and perform statistical operations. Their dual nature as both numerical and methodological tools broadens their use across various disciplines.