Problem 32 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$A=\left[\begin{array}{rr}2 & -5 \\\0 & 7\end{array}\right] \quad B=\left[\begin{array}{rrr}3 &\frac{1}{2} & 5 \\\1 & -1 & 3\end{array}\right] \quad C=\left[\begin{array}{rrr}2 & -\frac{5}{2} &0 \\\0 & 2 & -3\end{array}\right]$$ $$D=\left[\begin{array}{lll}7 & 3\end{array}\right] \quad E=\left[\begin{array}{l}1 \\\2 \\\0\end{array}\right] \quad F=\left[\begin{array}{lll}1 & 0 & 0 \\\0 & 1 & 0 \\\0 & 0 & 1\end{array}\right]$$ $$G=\left[\begin{array}{rrr}5 & -3 & 10 \\\6 & 1 & 0 \\\\-5 & 2 & 2\end{array}\right] \quadH=\left[\begin{array}{rr}3 & 1 \\\2 & -1\end{array}\right]$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. (a) \(A^{2}\) (b) \(F^{2}\)

Step-by-Step Solution

Answer

(a) \(A^2 = \begin{bmatrix} 4 & -45 \\ 0 & 49 \end{bmatrix}\). (b) \(F^2 = F\), the identity.

Step 1: Determine Dimensions for A^2

Matrix multiplication is only possible if the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix \(A\) is a 2x2 matrix. When squaring \(A\) (i.e., \(A \times A\)), we'll have two matrices of dimensions 2x2 each. Therefore, \(A^2\) can be computed.

Step 2: Calculate A^2

To calculate \(A^2\), we compute the product \(A \times A\). Using the entries of \(A\):\[A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]Multiply them:\[A \times A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}= \begin{bmatrix} (2\times2) + (-5\times0) & (2\times-5) + (-5\times7) \ (0\times2) + (7\times0) & (0\times-5) + (7\times7) \end{bmatrix}= \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix}.\]

Step 3: Determine Dimensions for F^2

Matrix \(F\) is a 3x3 identity matrix. When squaring \(F\) (i.e., \(F \times F\)), we will multiply two 3x3 matrices, which is possible since the number of columns of the first matrix is equal to the number of rows of the second.

Step 4: Calculate F^2

The identity matrix \(F\) is already a matrix that, when multiplied with itself, will yield itself:\[F = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]Multiplying \(F\) by itself, we find:\[F \times F = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]So, \(F^2 = F\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves two matrices and results in a new matrix. The process requires that the number of columns in the first matrix be equal to the number of rows in the second matrix. For example, if you have a matrix \( A \) of dimensions \( m \times n \) and a matrix \( B \) of dimensions \( n \times p \), the multiplication \( A \times B \) will produce a new matrix of dimensions \( m \times p \).

When multiplying matrices, each element of the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. This step involves multiplying corresponding elements and summing these products.

If this condition is not met, matrix multiplication cannot occur. For instance, when squaring a matrix like \( A^2 \), it must be a square matrix, as in matrix \( A \), which is \( 2 \times 2 \). This ensures that the multiplication can proceed as the numbers of columns and rows align properly.

Identity Matrix

The identity matrix is a special kind of square matrix that acts like the number 1 in matrix algebra. For any matrix \( M \), when multiplied by the identity matrix, the result is \( M \) itself, akin to how multiplying any number by 1 results in the original number.

An identity matrix is denoted as \( I \), and its main diagonal (the line of entries starting from the top left to the bottom right) is filled with ones, while all other elements are zeroes. For a 3x3 matrix, the identity matrix looks like this:

• \[\begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix}\]

In the exercise, matrix \( F \) is a 3x3 identity matrix. Multiplying the identity matrix by itself or by any other compatible matrix results in the same identity matrix or the original matrix, respectively. This is why \( F^2 = F \).

Understanding identity matrices is crucial as they serve as a foundational concept in linear transformations and matrix decompositions.

Square Matrices

Square matrices are matrices that have the same number of rows and columns (\( n \times n \)). They are pivotal in linear algebra as they allow certain operations such as finding determinants, inverses, and eigenvalues which are not applicable to non-square matrices.

In many mathematical contexts, square matrices have intriguing properties. For example:

• The product of a square matrix and its inverse (if it exists) is the identity matrix.
• Square matrices can be diagonalized, which simplifies many operations like exponentiation.

In the context of the given problems, both \( A \) and \( F \) are square matrices, which is significant because it allows these matrices to be squared, i.e., multiplied by themselves. The property of being a square matrix ensures the structural compatibility for operations like squaring, which involves multiplying a matrix by itself.