Problem 39 - Algebra and Trigonometry

Question

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ B C+C B $$

Step-by-Step Solution

Answer

The sum of matrices BC and CB is \[ \begin{bmatrix} 7 & 1 \ -7 & -1 \end{bmatrix} \]

Step 1: Multiplication of Matrices B and C

To perform the matrix multiplication operation BC, multiply each element of a row in the first matrix(B) by the corresponding element in a column of the second matrix(C) and then add those values together. For instance, to find element (1,1) of Matrix BC is obtained by multiplying element (1,1) in Matrix B with element (1,1) in Matrix C and adding it to the product of element (1,2) in Matrix B and element (2,1) in Matrix C. Follow this method for all the elements. Here, BC results in the following matrix: \[ \begin{bmatrix} 4 & 4 \ -4 & -4 \end{bmatrix} \]

Step 2: Multiplication of Matrices C and B

Follow the same matrix multiplication operation for CB, keeping in mind that matrix multiplication is not commutative, so BC and CB are not necessarily going to be equal. Recalculate the product using Matrix C first and Matrix B second. The result is: \[ \begin{bmatrix} 3 & -3 \ -3 & 3 \end{bmatrix} \]

Step 3: Sum of Matrices BC and CB

Add the values of corresponding elements in matrices BC and CB. The resulting matrix of BC + CB will be: \[ \begin{bmatrix} 7 & 1 \ -7 & -1 \end{bmatrix} \]

Key Concepts

Matrix Operations

Matrix operations are fundamental in linear algebra and play a critical role in various mathematical and practical applications.

When working with matrices, we can perform operations such as addition, subtraction, and multiplication. Each of these operations follows specific rulesets that dictate how to combine the matrices in order to produce a new matrix.

• For matrix addition, add corresponding elements of the matrices together. The two matrices must be of the same dimensions.
• Matrix multiplication, on the other hand, involves a row-by-element multiplication between a row of the first matrix and a column of the second matrix, followed by a sum of these products.
• Scalar multiplication entails multiplying every element of a matrix by a single number, known as a scalar.

It's crucial to remember that the order of multiplication matters and that some matrices cannot be multiplied if their dimensions are incompatible.

Non-Commutative Property

Unlike regular number multiplication, matrix multiplication does not adhere to the commutative property.

This means that the order in which you multiply two matrices can affect the outcome. In mathematical terms, for two matrices, A and B, it is not always the case that AB equals BA.

When it comes to the non-commutative nature of matrices, it is vital to always check the order of your factors. As illustrated in the solution provided, multiplying matrix B by matrix C yields a different result than multiplying matrix C by matrix B, demonstrating the non-commutative property of matrix multiplication.

Understanding this concept is essential for avoiding errors in calculations that involve sequences of matrix multiplications.

Matrix Addition

Matrix addition is one of the more straightforward operations within linear algebra. For two matrices to be added together, they must share the same dimensions, meaning they have the same number of rows and columns.

To perform matrix addition, simply add the corresponding elements from each matrix. For example, the element in the first row, first column of Matrix A is added to the element in the first row, first column of Matrix B, and so on for all other elements in the matrices.

The step-by-step solution provided showcases matrix addition by adding together BC and CB from the exercise. This addition results in each corresponding entry of the two matrices being summed individually to form the elements of a new matrix.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It lays the foundation for many areas of mathematics and applications in science and engineering, including computer graphics, optimization, and systems theory.

At its core, linear algebra involves the study of lines and planes, and the ways they can be mapped and transformed through matrices.

Matrix operations like multiplication and addition are the building blocks of linear algebra, and understanding these can help to unravel more complex problems involving vector spaces and linear mappings.

In the context of the given exercise, linear algebra principles helped in the computations of BC, CB, and their subsequent addition, illustrating how matrix operations are essential tools in the study and application of linear algebra.