Problem 36 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{rr}{-10} & {20} \\ {5} & {25}\end{array}\right], B=\left[\begin{array}{rr}{40} & {10} \\ {-20} & {30}\end{array}\right], C=\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1} \\ {1} & {0}\end{array}\right] $$ $$ C^{2} $$

Step-by-Step Solution

Answer

Matrix \( C^2 \) cannot be calculated because \( C \) is not square.

Step 1: Understanding the Matrix C

The matrix \( C \) is a \( 3 \times 2 \) matrix, meaning it has 3 rows and 2 columns. It is given by: \[ C = \begin{bmatrix} -1 & 0 \ 0 & -1 \ 1 & 0 \end{bmatrix} \] This means that the elements are arranged with three rows and two columns.

Step 2: Matrix Squaring Condition

To find \( C^2 \), we need to multiply matrix \( C \) by itself, i.e., \( C \cdot C \). However, for two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, \( C \) is \( 3 \times 2 \), meaning it has 2 columns, but it must be multiplied by another matrix that has 2 rows, which is not possible with \( C \).

Step 3: Identifying the Matrix Dimension Mismatch

Since the matrix \( C \) for multiplication is \( 3 \times 2 \) (3 rows and 2 columns), and we are supposed to multiply another \( 3 \times 2 \) matrix (since we are multiplying \( C \) by itself), the multiplication cannot be performed due to a dimension mismatch (the resultant matrix should have columns equal to rows of \( C \), which is not the case).

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial in understanding and performing matrix operations. Each matrix is defined by its number of rows and columns, commonly expressed as 'rows x columns' or \( m \times n \).
For instance, in our example, matrix \( C \) is a \( 3 \times 2 \) matrix because it has 3 rows and 2 columns.

• The first number represents the rows, the horizontal lines of elements across the matrix.
• The second number represents the columns, the vertical lines of elements down the matrix.

It's essential to understand these dimensions before attempting operations like multiplication. It ensures you know what other matrices are compatible for operations. For two matrices to multiply successfully, the number of columns in the first matrix must match the number of rows in the second matrix.

Matrix Operations

Matrix operations include addition, subtraction, and multiplication. Each has specific rules:
**Addition & Subtraction:** These operations require matrices to have the same dimensions. You simply add or subtract corresponding elements.
**Multiplication:** Matrix multiplication is more involved and requires specific dimensional compatibility:

• The number of columns in the first matrix must equal the number of rows in the second for multiplication to be possible.
• The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second.

This compatibility issue is why multiplying matrix \( C \) by itself isn't possible because \( C \) is a \( 3 \times 2 \) matrix. Attempting to multiply it by another \( 3 \times 2 \) matrix fails dimensionally, as 2 (columns of \( C \)) does not equal 3 (rows of the second \( C \)). This mismatch prevents multiplication.

Matrix Squaring

Matrix squaring involves multiplying a matrix by itself. For this operation, the matrix must be a square matrix, meaning it has the same number of rows and columns (e.g., \( 2 \times 2 \), \( 3 \times 3 \)).
When a matrix is square, the product remains compatible dimensionally:

• For example, a \( 2 \times 2 \) matrix multiplied by itself will produce another \( 2 \times 2 \) matrix.
• This is because the product of multiplication between respective rows of the first matrix and columns of the second will yield matching dimensions.

In our exercise, the challenge was to find \( C^2 \). However, \( C \) isn't square because it is \( 3 \times 2 \), making matrix squaring impossible. Understanding this necessity can help easily determine whether matrix squaring or other operations can proceed.