Problem 26 - 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. $$ B C $$

Step-by-Step Solution

Answer

The multiplication \( BC \) cannot be performed, as their dimensions are incompatible.

Step 1: Determine matrix dimensions

Matrix \( B \) has dimensions \( 2 \times 3 \) and matrix \( C \) has dimensions \( 2 \times 3 \). When multiplying matrices, the number of columns in the first matrix must match the number of rows in the second matrix.

Step 2: Verify multiplication compatibility

To multiply \( B \) and \( C \), we need the number of columns in \( B \) (which is 3) to match the number of rows in \( C \) (which is 2). Since this is not the case, the multiplication \( BC \) cannot be performed.

Key Concepts

Matrix Multiplication

Matrix Multiplication is a critical operation in linear algebra where two matrices are combined to produce another matrix. The key to understanding this operation is knowing how the elements from one matrix interact with the elements from another. In essence, each element of the resulting matrix is the sum of the products of corresponding entries from rows of the first matrix and columns of the second matrix. This might sound complex, but with practice, it becomes more intuitive.

Let's break it down further:

• The element in the position \(i,j\) of the resulting matrix is calculated by taking the dot product of the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix.
• This involves multiplying each element of the row by the corresponding element of the column, and then summing these products together.

Remember, only when the matrices are compatible (explained later) do we proceed with matrix multiplication. Otherwise, like in this exercise, it cannot be performed.

Matrix Dimensions

Matrix dimensions are indicative of the "size" or "shape" of a matrix, defining how many rows and columns it contains. A matrix with *m* rows and *n* columns is referred to as a \(m \times n\) matrix.

Knowing the dimensions is crucial when dealing with matrices because it tells us whether certain operations can be performed:

• For matrix addition, both matrices need to have the same dimensions, meaning the same number of rows and columns.
• For multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. This rule is vital for checking compatibility (as explored further in the next section).

The example given in the exercise helps emphasize how essential it is to understand dimensions; here, both matrices \( B \) and \( C \) were \( 2 \times 3\), thus preventing their multiplication.

Matrix Compatibility

Matrix Compatibility primarily concerns whether a desired matrix operation, particularly multiplication, can be performed. For two matrices to be compatible for multiplication, specific criteria must be fulfilled based on their dimensions.

Here’s what you need to know:

• To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
• If this condition isn't met, like with matrices \( B \) and \( C \) in the exercise, multiplication isn’t possible.

This makes checking compatibility a crucial step before attempting matrix operations. When compatible matrices are multiplied, the resulting matrix will have dimensions determined by the number of rows from the first matrix and the number of columns from the second matrix. Understanding these principles of matrix compatibility can help prevent errors and ensure efficient resolution of algebraic operations.