Problem 35 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ B^{2} $$

Step-by-Step Solution

Answer

The operation \(B^2\) cannot be performed as \(B\) is not a square matrix.

Step 1: Confirm the Matrices' Dimensions

Matrix \(B\) is a \(2 \times 3\) matrix, which means it has 2 rows and 3 columns.

Step 2: Understand Matrix Squaring

Squaring a matrix \(B\) means multiplying \(B\) by itself, i.e., \(B \cdot B\). This requires \(B\) to be a square matrix (the number of rows should equal the number of columns).

Step 3: Check Compatibility for Multiplication

For two matrices \(X\) and \(Y\) to be multiplied, the number of columns in \(X\) must equal the number of rows in \(Y\). For \(B^2\) (or \(B \cdot B\)), the number of columns in \(B\) (3 columns) does not match the number of rows in \(B\) (2 rows).

Step 4: Conclude the Operation's Impossibility

Since \(B\) is not a square matrix, we cannot multiply \(B\) by itself. Therefore, the operation \(B^2\) cannot be performed.

Key Concepts

Understanding Matrices

Matrices are rectangular arrays of numbers organized in rows and columns, which make them a powerful tool in linear algebra.
A matrix is a way to succinctly handle multiple numbers, variables, or equations at once.
Each entry in a matrix is called an element, and these elements are typically organized in brackets.

• Matrices can be used to perform various operations, such as addition, subtraction, and multiplication.
• They are used in a wide range of applications, from computer graphics to statistical analysis.

To fully understand matrices, it's important to first grasp their basic structure and properties, as well as the type of operations we can perform on them.

Matrix Dimensions

The dimensions of a matrix define its size and shape.
This is expressed in terms of the number of rows (horizontal lines) and columns (vertical lines) it has.
For example, a matrix with 2 rows and 3 columns is known as a 2 × 3 matrix.

• The dimensions of a matrix are critical in determining if certain algebraic operations can be carried out, such as multiplication.
• For multiplication, the number of columns in the first matrix must be the same as the number of rows in the second matrix.

This requirement leads to an important concept of compatibility, especially for operations like squaring a matrix, which can only be attempted if the matrix is square (same number of rows and columns).
In cases where these conditions aren't met, operations like multiplication cannot be performed.

Algebraic Operations with Matrices

Algebraic operations with matrices include addition, subtraction, and multiplication.
Each operation has its own set of rules based on the matrices involved. **Matrix Addition and Subtraction**

• Can only be performed on matrices of the same dimensions.
• Done element-wise, which means corresponding elements of the matrices are added or subtracted from each other.

**Matrix Multiplication**

• Involves a more complex process compared to addition or subtraction. The matrix dimensions must be compatible.
• When multiplying matrix A by matrix B, the number of columns in A must match the number of rows in B.

Matrix squaring is a special case where a matrix is multiplied by itself. However, this can only be done with square matrices, where the number of rows and columns are equal.
For instance, since matrix B has dimensions 2 × 3, it is not square and therefore cannot be squared, because the operation would require a 3 × 2 matrix.