Problem 20 - Precalculus

Question

Use the matrices \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}0 & -3 \\ -5 & 2\end{array}\right] \quad C=\left[\begin{array}{rrr}10 & -\frac{11}{2} & 0 \\ \frac{3}{5} & 5 & 9\end{array}\right]\) \(D=\left[\begin{array}{rr}7 & -13 \\ -\frac{4}{3} & 0 \\\ 6 & 8\end{array}\right] \quad E=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 4 & -9 \\ 0 & 0 & -5\end{array}\right]\) to compute the following or state that the indicated operation is undefined. $$ C D E $$

Step-by-Step Solution

Answer

The operation \(CDE\) is undefined because \(DE\) cannot be multiplied together.

Step 1: Check Dimensions

\(C\) is \(2 \times 3\), \(D\) is \(3 \times 2\), \(E\) is \(3 \times 3\).

Step 2: Evaluate CDE

We could compute \(CDE\) as \((CD)E\) or \(C(DE)\).
For \(DE\): \(D\) is \(3 \times 2\) and \(E\) is \(3 \times 3\). Since columns of \(D\) (2) \(\neq\) rows of \(E\) (3), \(DE\) is undefined.
For \(CD\): \(C\) is \(2 \times 3\) and \(D\) is \(3 \times 2\). This gives a \(2 \times 2\) matrix. Then \((CD)E\): \(2 \times 2\) times \(3 \times 3\) — columns of \(CD\) (2) \(\neq\) rows of \(E\) (3), so this is also undefined.

Step 3: State the Conclusion

The operation \(CDE\) is undefined because neither grouping \((CD)E\) nor \(C(DE)\) yields compatible dimensions for multiplication.

Key Concepts

Understanding Matrices

Matrices are essential components in the realm of linear algebra. A matrix is essentially a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The concept of matrices underlies many mathematical operations and has practical applications ranging from solving equations to computer graphics.

• Each entry in a matrix is known as an element. Elements are typically denoted by their position in the matrix, such as the entry in the second row, third column being represented as \(a_{23}\).
• Matrices are often categorized by their dimensions, which are defined by the number of rows and columns they possess. For example, a 2x3 matrix has 2 rows and 3 columns.

Understanding the arrangement and structure in matrices is crucial for conducting more complex operations, such as matrix multiplication.

Matrix Multiplicability

The rules governing whether two matrices can be multiplied hinge on a concept known as multiplicability. Matrix multiplication is not as straightforward as arithmetic multiplication of numbers.

For two matrices to be multiplicable,

• the number of columns in the first matrix must equal the number of rows in the second matrix.
• Once this condition is met, the resulting product matrix will have dimensions equal to the number of rows from the first matrix and the number of columns from the second matrix.

In our exercise, matrix \( C \) (2x3) can be multiplied by matrix \( D \) (3x2), leading to a product matrix with dimensions 2x2.

Matrix Dimensions

Matrix dimensions play a crucial role in determining whether certain matrix operations can be performed. The dimensions of a matrix are a direct indication of its shape and capacity in mathematical operations.

• A common notation to express matrix dimensions is to first state the number of rows followed by the number of columns, such as \(A\) being a 2x2 matrix if it has two rows and two columns.
• Understanding dimensions is vital when planning to perform operations like addition, subtraction, or multiplication, as these operations require matrices of compatible sizes.

Careful attention to dimensions can prevent undefined operations, like in our exercise, where the multiplication of matrices \(D\) (3x2) and \(E\) (3x3) failed because of a mismatch between the number of columns of \(D\) and the number of rows of \(E\).

Matrix Operations

Matrix operations extend the utility of matrices beyond simple representations of data. These operations include addition, subtraction, and multiplication, each with its own set of rules:

• Addition/Subtraction: These operations can only be performed on matrices of the same dimensions. Corresponding elements are added or subtracted.
• Multiplication: Unlike addition and subtraction, the multiplication of matrices is more complex and involves dot products. The elements of the resulting matrix are obtained by multiplying rows by columns across the two matrices.

Mastering basic matrix operations allows us to explore more advanced topics in mathematics, such as transformations and systems of equations. In practice, incorrect matrix operations can often lead to results being undefined, as demonstrated in our exercise where the operation \(CDE\) was declared undefined due to mismatched multiplication conditions.