Problem 35 - Algebra and Trigonometry

Question

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{rrrr} {2} & {-3} & {1} & {-1} \\ {1} & {1} & {-2} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {2} \\ {-1} & {1} \\ {5} & {4} \\ {10} & {5} \end{array}\right] $$

Step-by-Step Solution

Answer

Matrix AB is \(\left[\begin{array}{cc} -2 & 0 \ 6 & 2\end{array}\right]\). Matrix BA does not exist as the dimensions of the matrices do not permit multiplication in this order.

Step One: Analyzing the given matrices

The given matrices are:\nA=\(\left[\begin{array}{rrrr}{2} & {-3} & {1} & {-1} \ {1} & {1} & {-2} & {1}\end{array}\right]\) with dimensions 2x4.\nB=\(\left[\begin{array}{rr}{1} & {2} \ {-1} & {1} \ {5} & {4} \ {10} & {5}\end{array}\right]\) with dimensions 4x2.

Step Two: Verifying the multiplication feasibility

For AB, number of columns in A is equal to number of rows in B, i.e., 4=4. Therefore, AB multiplication is feasible.\nFor BA, number of columns in B is not equal to number of rows in A, i.e., 2!=2, so BA multiplication is not feasible.

Step Three: Calculating Matrix Multiplication

Only AB can be calculated. Using matrix multiplication rules yields:\nAB=\(\left[\begin{array}{cc}2*1-3*-1+1*5-1*10 & 2*2-3*1+1*4-1*5 \ 1*1+1*-1-2*5+1*10 & 1*2+1*1-2*4+1*5\end{array}\right]\)=\(\left[\begin{array}{cc} -2 & 0 \ 6 & 2\end{array}\right]\)

Step Four: Summarizing the results

The multiplication AB =\(\left[\begin{array}{cc} -2 & 0 \ 6 & 2\end{array}\right]\, and \, \) the multiplication BA is not feasible as the dimensions of the matrices do not permit multiplication in this order.

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is essential when dealing with matrix operations. Simply put, the dimensions of a matrix refer to the number of rows and columns that matrix contains. It’s customary to describe these dimensions as 'rows by columns'. For example, if we have a matrix A with dimensions 2x4, it means that matrix A has 2 rows and 4 columns.

When visualizing a matrix, imagine a grid where each cell contains a number or a variable. If we look at matrix B from the problem, which has dimensions 4x2, we can imagine a grid of four horizontal lines (rows) and two vertical lines (columns), yielding eight cells, each housing a specific value. It is the compatibility of these dimensions that determines whether two matrices can be multiplied, as explored in our next section.

Feasibility of Matrix Operations

The feasibility of matrix operations, particularly multiplication, hinges on a simple yet crucial rule: the number of columns in the first matrix must equal the number of rows in the second matrix. This condition ensures proper alignment of the elements when performing the multiplication.

For instance, if we take matrix A (2x4) and matrix B (4x2) from the exercise, we note that the number of columns in A matches the number of rows in B. This satisfies the rule, making the multiplication AB possible. Conversely, when attempting the product BA, we are faced with an incompatibility; the 2 columns of B do not match the 2 rows of A, thus precluding multiplication. Recognizing the feasibility of operations is fundamental before proceeding to any actual calculations to ensure correctness in mathematical processing.

Matrices in Algebra

Matrices play an instrumental role in algebra by facilitating the representation and manipulation of linear equations and transformations. They serve as a compact way to handle systems of equations, and their structured format allows us to perform a host of operations including addition, subtraction, multiplication, and finding inverses.

In algebraic terms, matrices simplify many problems by enabling operations on multiple equations concurrently. Matrix multiplication, for example, corresponds to composing linear transformations, a critical concept in higher-level mathematics and applications like computer graphics and data analysis. The procedure employs a dot product of the rows of the first matrix with the columns of the second matrix, a step that compels us to pay close attention to the earlier discussed dimensions for feasibility.

On a higher level, matrices are not just arrays of numbers but representations of abstract concepts and as such, they are integral components used in various fields such as physics, engineering, computer science, and economics to model and solve complex problems efficiently.