Problem 15 - Precalculus: Functions and Graphs

Question

Find, if possible, \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr} 4 & -3 & 1 \\ -5 & 2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 0 & 1 \\ -4 & 7 \end{array}\right]$$

Step-by-Step Solution

Answer

AB is \(\begin{bmatrix} 4 & 8 \\ -18 & 11 \end{bmatrix}\); BA is not possible.

Step 1: Determine the Dimensions

Matrix A is a 2x3 matrix, and Matrix B is a 3x2 matrix. To multiply two matrices, the number of columns of the first matrix must match the number of rows of the second matrix. Here, we can multiply A and B since the number of columns in A (3) equals the number of rows in B (3).

Step 2: Calculate Matrix AB

Matrix AB is obtained by multiplying matrix A (2x3) with matrix B (3x2). The result will be a 2x2 matrix. The element in the ith row and jth column of AB is obtained by multiplying the ith row of A by the jth column of B and summing the products.- For element (1,1): \(4 \times 2 + (-3) \times 0 + 1 \times (-4) = 8 + 0 - 4 = 4\)- For element (1,2): \(4 \times 1 + (-3) \times 1 + 1 \times 7 = 4 - 3 + 7 = 8\)- For element (2,1): \(-5 \times 2 + 2 \times 0 + 2 \times (-4) = -10 + 0 - 8 = -18\)- For element (2,2): \(-5 \times 1 + 2 \times 1 + 2 \times 7 = -5 + 2 + 14 = 11\)Thus, \(AB = \left[\begin{array}{rr} 4 & 8 \ -18 & 11 \end{array}\right]\).

Step 3: Consider the Possibility of BA

Matrix B (3x2) and Matrix A (2x3) cannot be multiplied to obtain BA because multiplication requires that the number of columns in B (2) must equal the number of rows in A (2). Here, the number of columns equals the number of rows, which makes BA possible. However, keeping checks in matrix dimensions, B (3x2) cannot be multiplied by A (2x3) as 2 is not equal to 3.

Step 4: Final Conclusion

The matrix multiplication AB is possible and equals \(\begin{bmatrix} 4 & 8 \ -18 & 11 \end{bmatrix}\). However, the multiplication BA is not valid due to dimension mismatch.

Key Concepts

Matrix Dimensions

When handling matrices, understanding the matrix dimensions is essential. The dimensions of a matrix describe its shape, written as 'rows x columns'. For example, a 2x3 matrix has 2 rows and 3 columns. In matrix multiplication, compatibility of matrix dimensions is critical: the number of columns in the first matrix must match the number of rows in the second matrix. This ensures that each row-element of the first matrix can align with each column-element of the second one, allowing for proper element-wise multiplication and summation.

Matrix Product

The matrix product is the result of multiplying two matrices together. When calculating the product of matrices A and B, given A is of dimensions p x q, and B is of dimensions q x r, the resulting product will be a new matrix with dimensions p x r. Each element in the resulting matrix is obtained by taking the dot product of the rows of the first matrix and the columns of the second matrix. Essentially, this means multiplying corresponding entries in row and column pairs and summing them up to get the final value for each element. This process is repeated for each element in the resulting matrix.

Matrix Operations

Matrix operations include a variety of calculations like addition, subtraction, and multiplication. Among these, multiplication is slightly more complex due to specific dimension requirements. Unlike addition and subtraction, which require matrices of identical dimensions, multiplication only necessitates matching the columns of the first matrix with the rows of the second one. These operations are fundamental in solving systems of equations, transforming geometry, and processing information in numerous fields, from computer graphics to data science. The key is understanding how these matrix operations transform data and how they are used in practical applications.

Elementary Matrix Operations

Elementary matrix operations are basic manipulations that are used in matrix algebra. They include operations such as row switching, row multiplication (by a non-zero scalar), and row addition/subtraction (adding/subtracting one row to/on another). These operations are commonly used to simplify matrices, especially in solving linear algebra problems like systems of linear equations. By applying these operations, matrices can be manipulated into simpler forms, such as row-echelon form or reduced row-echelon form, which makes it easier to analyze and solve them.