Problem 36 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to calculate the square of matrix C, denoted as \( C^2 \). This means multiplying matrix C by itself: \( C \times C \). Matrix \( C \) is a 3x2 matrix as it has 3 rows and 2 columns.

Step 2: Check Dimensions for Multiplication

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In this case, matrix \( C \) is 3x2, and to multiply it by itself, the second \( C \) needs to be 2x3 to make the operation possible. However, both matrices \( C \) are 3x2, hence the multiplication cannot be performed.

Step 3: Conclude the Result

Since the number of columns in the first \( C \) (which is 2) does not equal the number of rows in the second \( C \) (which is 3), the matrix multiplication \( C \times C \) is not possible.

Key Concepts

Matrix Operations

Matrix operations involve various mathematical procedures, with matrix multiplication being one of the most crucial ones. It is essential to note that not all operations are possible on all matrices at all times. Common matrix operations include:

• Matrix addition and subtraction: Works when matrices have identical dimensions.
• Matrix multiplication: Requires specific conditions to be met regarding dimensions.
• Scalar multiplication: A straightforward process of multiplying each entry by a scalar (a constant numerical value).

One key operation is matrix multiplication, which is not as intuitive as addition or subtraction. It isn't merely pairing corresponding elements like in addition or subtraction. Instead, you take the rows of the first matrix and the columns of the second matrix. Then, you multiply and sum corresponding elements. It's important to specify that for such multiplication to occur, certain conditions must be met, which leads us to the next core concept: matrix dimensions.

Matrix Dimensions

Understanding matrix dimensions is fundamental in determining if matrix operations, such as multiplication, can be performed. Matrix dimensions are described using the format 'rows x columns' (e.g., a 3x2 matrix has 3 rows and 2 columns). This designation helps identify what operations are possible.

For instance, when performing matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. This compatibility ensures each element in a resulting matrix is well-defined. If not, the multiplication operation is not defined, which brings us to the concept of non-conformable matrices.

Observing dimensions of a matrix helps anticipate issues like non-conformability and avoid errors in mathematical operations. Ensuring matrices are conformable before attempting operations is essential in mathematical computation.

Non-Conformable Matrices

Non-conformable matrices refer to matrices that do not meet the required condition for a specific operation—particularly multiplication. As seen in the given exercise with the matrix C:

• Matrix C is a 3x2 matrix.
• For multiplication with another matrix C (also 3x2), the columns of the first matrix must match the rows of the second, which is not the case here.
• Therefore, multiplying matrix C by itself is impossible as they don’t conform to the rules for multiplication.

These matrices highlight a critical aspect of matrix operations—dimensions play such an integral role that non-conformability means an operation simply cannot proceed. Understanding this helps in swiftly determining the feasibility of operations and ensures mathematical correctness.