Problem 28 - Algebra and Trigonometry

Question

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{rr} {3} & {-2} \\ {1} & {5} \end{array}\right], \quad B=\left[\begin{array}{rr} {0} & {0} \\ {5} & {-6} \end{array}\right] $$

Step-by-Step Solution

Answer

The product of the matrices AB is \( \left[\begin{array}{rr}{-10} & {12}\{25} & {-30} \end{array}\right] \) and the product of the matrices BA is \( \left[\begin{array}{rr}{0} & {0}\{9} & {-40}\end{array}\right] \)

Step 1: Multiply the matrices AB

To find \( AB \), perform the multiplication as follows:(3*0 + -2*5 , 3*0 + -2*-6)(1*0 + 5*5 , 1*0 + 5*-6)Resulting in \( AB = \left[\begin{array}{rr}{-10} & {12}\{25} & {-30} \end{array}\right] \)

Step 2: Multiply the matrices BA

To find \( BA \), perform the multiplication as follows:(0*3 + 0*1 , 0*-2 + 0*5 )(5*3 + -6*1 , 5*-2 + -6*5)Resulting in \( BA = \left[\begin{array}{rr}{0} & {0}\{9} & {-40}\end{array}\right] \)

Key Concepts

Matrix Operations

Matrix operations are a fundamental part of linear algebra. One common operation is matrix multiplication, which combines two matrices to form a new matrix. This process is essential in various fields like physics, engineering, and computer graphics.
Matrix multiplication is not as straightforward as multiplying single numbers. Instead, it involves the sum of products of the elements from the rows of the first matrix with columns of the second matrix. Each element in the resulting matrix is calculated by:

• Multiplying corresponding elements from a row of the first matrix by a column of the second matrix.
• Summing these products to get each element in the new matrix.

Importantly, matrix multiplication is not commutative, meaning that multiplying matrix A by matrix B does not generally yield the same result as multiplying B by A.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and the linear mappings between them. It involves the study of lines, planes, and subspaces, and is fundamental in both pure and applied mathematics.
At the heart of linear algebra are matrices and vectors. These mathematical objects are tools to express and solve linear equations. They also make it easier to model and perform calculations for complex systems, like those found in economics or computer science.
This specific exercise focuses on matrix multiplication, which is a critical operation in linear algebra. Matrix operations facilitate the solving of systems of linear equations, transforming matrices, and understanding geometric transformations.

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix multiplication. Matrix dimensions describe the number of rows and columns in a matrix, generally noted as "m x n" where "m" is the row count, and "n" is the column count.
The ability to multiply two matrices depends on their dimensions. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. For example, a 2 x 2 matrix can multiply another 2 x 2 matrix or a 2 x 3 matrix, but not a 3 x 2 matrix.
In this exercise, both matrices A and B are 2 x 2, allowing both products AB and BA to be calculated. This distinction is vital because even if the dimensions permit multiplication, the resulting matrices have different values, demonstrating the concept of non-commutativity in matrix multiplication.