Problem 45 - 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: \(\left.A^{2}=A \cdot A\right)\) \(A=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{rrr}-2 & 3 & 4 \\ -1 & 1 & -5\end{array}\right], C=\left[\begin{array}{rr}0.5 & 0.1 \\ 1 & 0.2 \\ -0.5 & 0.3\end{array}\right], D=\left[\begin{array}{rrr}1 & 0 & -1 \\ -6 & 7 & 5 \\\ 4 & 2 & 1\end{array}\right]\) \(D^{2}\)

Step-by-Step Solution

Answer

The resulting matrix D² is \([-3, -2, -2], [-28, 59, 46], [-4, 16, 7]\).

Step 1: Identify the operation

The operation to be performed is squaring matrix \(D\), which means we need to find \(D \cdot D\).

Step 2: Determine dimensions of D

The matrix \(D\) is a \(3 \times 3\) square matrix. Squaring a matrix is possible only if it is square, which is the case here.

Step 3: Perform matrix multiplication

Multiply matrix \(D\) by itself. Remember that matrix multiplication involves taking the dot product of rows of the first matrix with columns of the second matrix.\[ D \cdot D = \begin{bmatrix} 1 & 0 & -1 \ -6 & 7 & 5 \ 4 & 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & -1 \ -6 & 7 & 5 \ 4 & 2 & 1 \end{bmatrix} \]

Step 4: Calculate each element of D²

Compute each element of the resulting matrix by calculating the dot products:- \((1,1)\) position: \((1)(1) + (0)(-6) + (-1)(4) = 1 - 4 = -3\)- \((1,2)\) position: \((1)(0) + (0)(7) + (-1)(2) = 0 - 2 = -2\)- \((1,3)\) position: \((1)(-1) + (0)(5) + (-1)(1) = -1 - 1 = -2\)- \((2,1)\) position: \((-6)(1) + (7)(-6) + (5)(4) = -6 - 42 + 20 = -28\)- \((2,2)\) position: \((-6)(0) + (7)(7) + (5)(2) = 49 + 10 = 59\)- \((2,3)\) position: \((-6)(-1) + (7)(5) + (5)(1) = 6 + 35 + 5 = 46\)- \((3,1)\) position: \((4)(1) + (2)(-6) + (1)(4) = 4 - 12 + 4 = -4\)- \((3,2)\) position: \((4)(0) + (2)(7) + (1)(2) = 0 + 14 + 2 = 16\)- \((3,3)\) position: \((4)(-1) + (2)(5) + (1)(1) = -4 + 10 + 1 = 7\)

Key Concepts

Square Matrices

A square matrix is a special type of matrix where the number of rows and columns are equal. This uniformity gives these matrices a kind of balanced structure that is vital for certain mathematical operations.

Square matrices, such as identity matrices and diagonal matrices, have distinct properties that make them essential in calculations. One key property is their ability to be squared, as only square matrices can undergo an operation where the matrix is multiplied by itself.

If you imagine a square matrix as a grid; multiplying it by itself is akin to overlaying the same grid on top of itself repeatedly, a process that may lead to unique and insightful results depending on the values within the matrix.

Matrix Dimensions

Matrix dimensions refer to the number of rows and columns a matrix has, expressed as \(m \times n\). This order matters greatly because it determines the operations we can perform with them.

For example, determining if two matrices can even be multiplied together hinges upon the inner dimensions being the same: the number of columns in the first matrix must match the number of rows in the second. If these conditions aren't met, the matrices can't be multiplied.

In the context of our exercise, knowing that matrix \(D\) is a \(3 \times 3\) helps us understand immediately why squaring it (\(D^2\)) is possible, since square matrices handle multiplication by themselves seamlessly.

Dot Product

The dot product is an essential component of matrix multiplication, playing the central role in determining product elements. While it originates from vector mathematics, its principles seamlessly apply to matrices.

In matrix multiplication, to derive each element of the product, you take the dot product of a row from the first matrix with a column from the second. To calculate this, multiply corresponding elements and sum these products. For instance, the element in the upper left corner of the product matrix results from dot multiplying the first row of the first matrix by the first column of the second matrix.

This operation may seem like small individual calculations, but together, they build the entire result of multiplying two matrices. So, understanding the dot product is crucial for mastering matrix multiplication.

Matrix Operations

Matrix operations encompass a variety of techniques for manipulating matrices, including addition, subtraction, and multiplication. Each operation has its own unique rules and requirements.

In terms of matrix multiplication, the process involves taking the dot product of rows and columns, as previously discussed. However, unlike addition or subtraction, multiplication isn't commutative, meaning \(A \times B\) isn't necessarily equal to \(B \times A\).

Understanding how matrix operations work helps students solve complex equations, reverse transformations, and analyze systems of equations. Concepts like identity matrices and zero matrices often come into play, especially in linear algebra, where matrices define vector spaces and transformations.