Problem 36 - College Algebra

Question

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{rrrr} 2 & -1 & 3 & 2 \\ 1 & 0 & -2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} -1 & 2 \\ 1 & 1 \\ 3 & -4 \\ 6 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The result for AB is \[\left[\begin{array}{cc}16 & 8 \-11 & -6 \end{array}\right]\], but \(BA\) is undefined because the number of columns in first matrix does not equal the number of rows in second matrix.

Step 1: Calculate AB

Perform the matrix multiplication \(AB\). In this case, you will consider each element of the resultant matrix \(C\) as \(C_{ij} = A_i * B_j\). So you multiply the \(i\)th row of matrix \(A\) by the \(j\)th column of matrix \(B\) and sum the result. This is done for each element of \(C\). The result will be a 2x2 matrix.

Step 2: Calculate BA

Attempt to perform the matrix multiplication \(BA\). Here, the number of columns in \(B\) does not match the number of rows in \(A\). Hence, the multiplication \(BA\) is not possible due to these incompatible dimensions.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is a fundamental aspect of linear algebra, crucial for performing matrix operations, including multiplication. The dimensions of a matrix are expressed as 'rows by columns'. For example, if we refer to matrix A as a 2x4 matrix, it means matrix A has 2 rows and 4 columns. Similarly, matrix B being a 4x2 matrix describes B having 4 rows and 2 columns.

When preparing for matrix multiplication, it's important to note that the number of columns in the first matrix must be equal to the number of rows in the second matrix. This compatibility is key to successful matrix multiplication and central to understanding why some multiplications may not be feasible.

Matrix Operations

Matrix operations, including addition, subtraction, and multiplication, follow certain rules dictated by the sizes and shapes of the matrices involved. Focusing on multiplication, which combines two matrices to form a new matrix, we follow a dot product operation.

To multiply matrices, we calculate the dot product of rows from the first matrix with columns from the second matrix. The resulting matrix dimensions are determined by the number of rows in the first matrix and the number of columns in the second matrix. These operations are not only pivotal in pure mathematics but also in practical applications such as computer graphics, economics, and engineering.

Incompatible Matrix Sizes

In matrix multiplication, incompatible matrix sizes occur when the number of columns in the first matrix does not equal the number of rows in the second matrix. This situation prevents the multiplication from being performed as there is no way to properly align the elements for the dot product operation.

For instance, if we attempt to multiply a matrix with dimensions 2x4 (matrix A) by one with dimensions 4x2 (matrix B), this operation is valid. However, reversing the order to multiply a 4x2 matrix by a 2x4 matrix (BA), we face incompatible sizes which make the operation undefined. It is essential to recognize incompatible sizes to avoid computational errors in matrix operations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is a foundational area that not only mathematicians but scientists and engineers need to understand thoroughly.

Matrix multiplication is a core operation in linear algebra that supports the conceptual framework for various transforms and operations across multi-dimensional spaces. Mastering matrix multiplication and understanding when and why certain multiplications fail due to incompatible sizes, such as in the exercise above, is a key skill in linear algebra. This knowledge forms the basis for more advanced concepts and applications of math in real-world scenarios.