Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{rrr} -1 & 2 & 3 \\ 4 & 1 & -1 \end{array}\right], B=\left[\begin{array}{rr} 1 & 3 \\ -1 & -2 \\ 2 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

(a) \(A B\) = \(\left[\begin{array}{rr}-4 & -4 \5 & 16\end{array}\right]\), (b) \(B A\) = \(\left[\begin{array}{rrr}11 & 5 & 1 \-3 & -3 & -1 \10 & 10 & 2\end{array}\right]\), (c) \(A^{2}\) is not possible to calculate because \(A\) is not a square matrix.

Step 1: Calculate AB

Multiply matrices \(A\) and \(B\). The result of \(A B\) = \(\left[\begin{array}{rr}-4 & -4 \5 & 16\end{array}\right]\) is obtained by summing products of elements in corresponding rows of matrix \(A\) and columns of matrix \(B\). The first row and first column products are ((-1)*1) + (2*(-1)) + (3*2) = -4. Similarly, the rest entries of the resulting matrix are calculated.

Step 2: Calculate BA

Next, multiply matrices \(B\) and \(A\). The result of \(B A\) = \(\left[\begin{array}{rrr}11 & 5 & 1 \-3 & -3 & -1 \10 & 10 & 2\end{array}\right]\) is obtained in a similar way by summing the products of corresponding rows of matrix \(B\) and columns of matrix \(A\).

Step 3: Calculate \(A^{2}\)

In a final step, you attempt to calculate \(A^{2}\). As mentioned in the analysis, the given matrix \(A\) is not a square matrix (it does not have the same number of rows as columns). Thus, the square of \(A\) is not defined and cannot be calculated.

Key Concepts

Matrix Operations

Matrix operations are foundational to linear algebra and have widespread applications in mathematics, physics, computer science, and more. The most common operations include addition, scalar multiplication, and perhaps one of the most critical, matrix multiplication. In the case of matrix multiplication, it involves taking two matrices, say matrix A and matrix B, and producing a new matrix which represents the composite of the linear transformations that each matrix represents.

However, it's crucial to acknowledge that matrices must follow certain rules to be eligible for multiplication. Specifically, the number of columns in the first matrix must be equal to the number of rows in the second. If matrix A has dimensions of mxn and matrix B has dimensions of nxp, then their product AB will have dimensions of mxp. The procedure includes multiplying the respective elements of rows of the first matrix with the columns of the second one and summing up their products to form a new entry in the product matrix.

Non-Square Matrix

Matrices can come in various shapes and sizes. Most people are familiar with square matrices, where the number of rows and columns are equal, but non-square matrices are equally important. A non-square matrix is simply a matrix that doesn't have the same number of rows and columns, such as a 2x3 or 4x1 matrix.

These matrices can still be used in many operations, but some, such as finding the determinant or the square of the matrix, are not applicable. This was the issue we encountered in our exercise with matrix A. As it is a 2x3 matrix, we cannot find its square, A², because matrix multiplication requires the inner dimensions to match. In simple terms, a non-square matrix cannot be multiplied by itself, as the multiplication operation is not defined in this context.

Matrix Algebra

Matrix algebra extends the concepts of algebra to matrices, encompassing the rules and operations that can be performed on matrices. While some of these mirror regular algebra, like distributive and associative properties, others are unique to matrix algebra, such as the non-commutative nature of matrix multiplication.

In our exercise, we explored whether AB is the same as BA. In general algebra, the multiplication of two numbers is commutative, meaning their order doesn't affect the outcome. However, in matrix algebra, AB may not equal BA, which our solution showcased. This non-commutative aspect highlights the importance of the order in which matrices are multiplied, as it can greatly impact the result.

Matrix algebra is also rich with other particularities, such as the existence of an identity matrix, which acts as the '1' in matrix multiplication, and the concept of inverse matrices, which functions similarly to division in regular algebra. Understanding the breadth of matrix algebra is crucial for solving advanced problems in various scientific and engineering fields.