Problem 36 - College Algebra with Corequisite Support

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 multiplication \( C^2 \) is not possible due to mismatched dimensions.

Step 1: Verify matrix dimensions

Matrix \( C \) is a \( 3 \times 2 \) matrix (3 rows and 2 columns). According to matrix multiplication rules, a \( m \times n \) matrix can multiply an \( n \times p \) matrix. For \( C^2 = C \cdot C \), both matrices here are \( 3 \times 2 \), so \( C \cdot C \) is not a valid operation because the number of columns in the first matrix (2) does not equal the number of rows in the second matrix (3).

Step 2: Conclusion on matrix multiplication possibility

Since the dimensions do not match the required format for matrix multiplication, \( C^2 \) is not defined. Thus, we cannot compute \( C \cdot C \).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in determining whether certain operations can be performed. The dimensions of a matrix tell us how many rows and columns it has. For example, a matrix that has 3 rows and 2 columns is referred to as a \( 3 \times 2 \) matrix. The dimensions are always written in the form of "rows \( \times \) columns". This layout helps us know how two matrices can "fit" together during operations like multiplication.

When multiplying two matrices, matrix dimensions play a key role. The rule for multiplying two matrices, say matrix A and matrix B, is that the number of columns in matrix A must equal the number of rows in matrix B. If matrix A is of size \( m \times n \) and matrix B is \( n \times p \), the resulting matrix will have the dimensions \( m \times p \).

• If the dimensions of the matrices do not align in this way, their multiplication is not possible.
• For square matrices (e.g., \( 2 \times 2 \), \( 3 \times 3 \)), multiplication is always defined, because the number of rows equals the number of columns.

Matrix Operations

Matrix operations include addition, subtraction, and multiplication. Each operation follows specific rules based on the matrix dimensions.

1. **Matrix Addition and Subtraction**: These operations require that the matrices have the same dimensions. With this requirement, each element in one matrix can be directly added to or subtracted from the corresponding element in the other matrix. If the dimensions are not equal, these operations cannot occur.

2. **Matrix Multiplication**: For two matrices to be multiplied, the columns in the first must match the rows in the second. When matrices are multiplied, each element of the resulting matrix is computed as the sum of products of elements from the rows of the first matrix and columns of the second matrix. This operation is more complex and is not commutative, meaning \( A \times B eq B \times A \).

• Unlike addition and subtraction, multiplication results in a new matrix whose dimensions depend on the input matrices' dimensions.
• Understanding matrix multiplication rules is fundamental in fields such as computer graphics, physics, and other applied sciences.

Linear Algebra

Linear Algebra is a branch of mathematics that focuses heavily on matrices and operations involving them. It is the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations.

Matrices are foundational in Linear Algebra because they represent transformations and linear equations. Within this field, understanding how to manipulate and operate with matrices is essential. The rules of operations, like matrix multiplication, factor into solving linear systems represented in matrix forms. For instance, finding solutions to multiple linear equations can often be visualized or calculated using matrices and their operations.

• Transformations, particularly in physics and engineering, are often expressed by matrices describing rotations, translations, and scaling.
• In Linear Algebra, the study extends beyond basic operations to include finding determinants, inverses, and understanding eigenvalues and eigenvectors.

Linear Algebra has profound implications in several scientific and engineering domains, offering tools for modeling and problem-solving in complex scenarios.