Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$d A B^{2}$$

Step-by-Step Solution

Answer

This operation is undefined because the square of matrix \( B \) is not possible. The number of columns in matrix \( B \) does not match the number of rows when attempting to square it, making the operation invalid.

Step 1: Calculate Square of B

The square of matrix \( B \) must be calculated first to obtain \( B^{2} \). Matrix \( B \) is a 2x3 matrix and squaring it is impossible because the number of columns in the first matrix (2) is not equal to the number of rows in the second matrix (3). Therefore, the operation \( B^{2} \) is undefined.

Step 2: Skip Multiplication of Matrices A and B

Since Step 1 cannot be performed, we cannot proceed to this step as this step needs the result of \( B^{2} \) to multiply with matrix \( A \). Therefore, the operation \( A \cdot B^{2} \) is also undefined.

Step 3: Omit Scalar Multiplication

The scalar multiplication, \( d \cdot A \cdot B^{2} \), involves multiplying the scalar \( d \) to the result of \( A \cdot B^{2} \). However, since the multiplication of matrices \( A \) and \( B^{2} \) is not valid, we cannot execute this step. Therefore, the expression \( d \cdot A \cdot B^{2} \) is also undefined.

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. It involves taking two matrices and producing a third matrix from them. The process of multiplying matrices is not as simple as multiplying their individual elements; instead, it follows specific rules. To successfully multiply a matrix by another matrix, the number of columns in the first matrix must match the number of rows in the second matrix. This matching dimension ensures that each element of the resulting matrix is calculated as the dot product of the corresponding row of the first matrix and the column of the second matrix. If the dimensional criteria aren't met, the matrix multiplication is not defined, leading to an undefined operation as seen in the problem where the square of matrix \( B \) attempts to multiply a 2x3 matrix by itself, quicly revealing this dimension mismatch. 

• Ensure correct matching of dimensions during multiplication.
• Resultant matrix dimensions: rows of the first matrix and columns of the second matrix.

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar (a constant). This operation is much simpler than matrix multiplication as there are no strict dimensional requirements. You simply multiply every entry within the matrix by the scalar. In this case, the scalar \( d = -3 \) wants to multiply matrix \( A \) which would mean that each element in matrix \( A \) would need to be multiplied by \(-3\). However, since the entire expression involved \( d \cdot A \cdot B^2 \) where \( A \cdot B^2 \) was undefined, the scalar multiplication was thus omitted. 

• Multiply each matrix element by the scalar.
• The operation doesn't alter the matrix structure, only the values.

Undefined Operations

Undefined operations occur when attempted operations do not meet necessary mathematical conditions to proceed. In the realm of matrices, this is frequently due to incompatible dimensions where multiplication is concerned. As shown in the exercise, squaring matrix \( B \) fails due to its 2x3 dimension, making it impossible to multiply it by itself because the number of columns in the first duplicate does not match the number of rows in the second duplicate. When an initial operation is undefined, all successive calculations relying on it also become undefined, as seen with \( A \cdot B^2 \) and \( d \cdot A \cdot B^2 \). Understanding why operations are undefined is crucial, as it highlights the need to carefully consider dimension requirements and order of operations whenever working with matrices. 

• Check dimensions before attempting matrix multiplication.
• Recognize that one undefined operation can affect subsequent calculations.

Graphing Utility

Graphing utilities are technological tools like graphing calculators or software that help perform complex mathematical operations with ease. They are especially valuable when dealing with matrices as they can quickly calculate results, verify correctness, or demonstrate why operations might be undefined through visual representation. Using a graphing utility can simplify the work when manually calculating such as multiplication of larger matrices can become tedious. They can also provide visual insights, such as graphing the transformation of spaces performed by matrices. However, it's important to remember that while graphing utilities handle computations, understanding the mathematics behind these operations ensures the user fully grasps why certain operations, such as those involving incompatible dimensions, may be undefined. 

• Use graphing utilities to save time on calculations.
• Verify the correctness of your manual computations.
• Gain better insights with visual representations.