Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Matrix Multiplication Use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$A=\left[\begin{array}{rrrr} 7 & 6 & 9 & -4 \\ 3 & -4 & 11 & -2 \\ -5 & -8 & 1 & 12 \end{array}\right], \quad B=\left[\begin{array}{rr} 15 & 8 \\ 23 & -17 \\ 9 & 10 \end{array}\right]$$

Step-by-Step Solution

Answer

The multiplication of matrix A and B is not possible since the number of columns of matrix A is not equal to the number of rows of matrix B.

Step 1: Analyze the matrices dimensions

Matrix A is a 3 by 4 matrix, meaning it has 3 rows and 4 columns. Matrix B is a 3 by 2 matrix, meaning it has 3 rows and 2 columns.

Step 2: Determine if the multiplication is possible

For matrix multiplication to occur, the number of columns in the first matrix must equal the number of rows in the second matrix. In this case, matrix A has 4 columns and matrix B has 3 rows. Therefore, the multiplication of matrix A and matrix B is not defined.

Step 3: Conclusion

Since the number of columns of matrix A is not equal to the number of rows of matrix B, the multiplication of the two matrices is not possible. This means that \( AB \) can't be computed.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when dealing with matrix multiplication. The dimensions of a matrix refer to the number of rows and columns it contains. This is typically noted as 'm x n', where 'm' represents the rows, and 'n' represents the columns.

For instance, in the given exercise, the matrix A has dimensions of 3x4, which means it has 3 rows and 4 columns. Matrix B has dimensions of 3x2, indicating 3 rows and 2 columns. The key to determining whether two matrices can be multiplied is in their dimensions; specifically, the inner dimensions must match. That is, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

The exercise above illustrates a common pitfall: assuming any two matrices can be multiplied. However, since matrix A's columns do not match matrix B's rows, the multiplication is impossible. It's essential for students to examine matrix dimensions carefully before attempting any operations.

Graphing Utility

A graphing utility can be an efficient tool in performing complex matrix operations, including multiplication. Such utilities often support matrix calculations, enabling students to input matrices and carry out operations with a few clicks or commands.

When using a graphing utility, you would typically start by entering the matrices into the system. It's important to ensure that each element of the matrix is entered correctly to prevent calculation errors. After the matrices are inputted, the utility can quickly compute products and other matrix-related operations, provided the matrices are compatible for the operation (as dictated by their dimensions).

Graphing utilities save time and reduce the risk of manual computation errors, making them a popular choice for students. However, one should still understand the underlying principles of matrix operations, as this knowledge is key to troubleshooting any issues that might arise when utilizing such tools.

Matrix Operations

Matrix operations are foundational in various fields, including mathematics, engineering, and computer science. They are used to solve systems of equations, transform geometric shapes, and represent data. Beyond multiplication, common matrix operations include addition, subtraction, and finding inverses or determinants.

Adding and Subtracting Matrices

These operations are only defined for matrices of the same dimensions. Matrices are added or subtracted element-wise, meaning each element in one matrix is added or subtracted from the corresponding element in the other matrix.

Matrix Inversion

Used for solving systems of linear equations, the inverse of a matrix effectively undoes its operations. However, not all matrices have inverses. A matrix must be square (same number of rows and columns) and non-singular – meaning it has a non-zero determinant.

Understanding these operations requires practice and a good grasp of the rules and conditions governing them. For instance, matrix multiplication is associative and distributive but not commutative. Knowing these properties facilitates more advanced studies and applications of matrices.