Problem 22 - 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. $$ C-5 A $$

Step-by-Step Solution

Answer

\( C - 5A \) cannot be performed as the matrices have different dimensions.

Step 1: Identify Matrix Dimensions

Matrix \( A \) is a 2x2 matrix, and matrix \( C \) is a 2x3 matrix. To perform the operation \( C - 5A \), both matrices must have the same dimensions.

Step 2: Check for Dimension Compatibility

Since matrix \( A \) is a 2x2 matrix and matrix \( C \) is a 2x3 matrix, they do not have the same dimensions. Matrix subtraction requires matrices to have the same dimensions.

Step 3: Determine Feasibility of Operation

Because \( A \) and \( C \) have different dimensions (\( A \) is 2x2 and \( C \) is 2x3), the subtraction operation \( C - 5A \) cannot be performed. Two matrices can only be added or subtracted if they have the same dimensions.

Key Concepts

Understanding Matrix Subtraction

Matrix subtraction is an essential operation that requires both matrices to have identical dimensions. This means a matrix with the same number of rows and columns must be involved for subtraction to be possible. For example, if you have a matrix like \( A \) which is 2x2, it can only be subtracted from another 2x2 matrix. If matrix \( C \) is 2x3, then subtraction cannot occur with \( A \).
Subtraction works element-wise, meaning each corresponding element from the two matrices is subtracted from one another.

• Each pair of corresponding elements (same position in each matrix) is operated upon.
• The resultant matrix is the same size as the initial matrices.

When the dimensions do not match, the operation is not feasible, and an error will result. Always verify dimensions before proceeding with subtraction.

Matrix Dimensions Explained

Matrices have dimensions that are defined by the number of rows and columns they possess. These dimensions are crucial when determining whether you can engage in specific operations like addition or subtraction.
For example, a matrix labeled as 2x2 has 2 rows and 2 columns, making it a square matrix. On the other hand, a matrix that is 2x3 has 2 rows and 3 columns, categorizing it as a rectangular matrix.

• Matching dimensions: Operations like subtraction can only happen when the matrices have identical dimensions.
• Mismatch dimensions: Operations are not feasible if matrix dimensions differ, as was the case in the example with matrices \( A \) and \( C \).

Always pay attention to the dimensions when working with matrices to avoid operational errors.

Algebraic Operations with Matrices

Algebraic operations on matrices include addition, subtraction, scalar multiplication, and more. These operations have specific rules that ensure they are conducted correctly.
Subtraction, for instance, involves a one-to-one correspondence between elements in matrices. Scalar multiplication, seen in the operation \( 5A \), involves multiplying each element of a matrix by a constant:

• Scalar Multiplication: An operation where every element of a matrix is multiplied by the same scalar. The size of the matrix stays the same.
• Subtraction: This requires the matrices to have equal dimensions to perform element-wise subtraction as described earlier.

Understanding these rules helps prevent mistakes in matrix operations and ensures that results are accurate.