Problem 36 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The result of \( F^2 \) is the identity matrix \( F \).

Step 1: Identify the Operation

The exercise requires computing the square of matrix \( F \), denoted as \( F^2 \). This implies multiplying matrix \( F \) by itself (\( F imes F \)).

Step 2: Verify Matrix Dimensions for Multiplication

Matrix multiplication requires that the number of columns in the first matrix must match the number of rows in the second matrix. Matrix \( F \) is a \( 3 \times 3 \) identity matrix, thus it can be multiplied by itself since both matrices are \( 3 \times 3 \).

Step 3: Perform Matrix Multiplication

Multiply each element of a row of the first matrix by the corresponding element of a column in the second matrix and sum the products to get each element of the product matrix. For the identity matrix \( F \):\[F^2 = F \times F = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]

Step 4: Verify the Result

The product of \( F \times F \) results in \( F \) because it's an identity matrix, and identity matrices serve as multiplicative identities in matrix algebra, i.e., any matrix multiplied by the identity matrix remains unchanged.

Key Concepts

Identity Matrix

An identity matrix is like the number 1 in regular multiplication, but for matrices. It "keeps" the other matrices the same when involved in multiplication. The identity matrix is square, meaning it has the same number of rows and columns, and is full of zeros except for a line of 1's running from the top left to the bottom right corner.

• This main diagonal of 1's is what makes it so special in matrix multiplication.
• Identity matrices can come in any size, but a common type is the 3x3, like matrix F in the original exercise.
• When you multiply any matrix by an identity matrix, it's as if nothing happened. The product is just the original matrix itself.

To put it simply, multiplying by an identity matrix doesn't "change" the other matrix. This makes it very powerful, especially when working with bigger matrix equations in mathematics.

Matrix Algebra

Matrix algebra is the study of how matrices can interact with one another through operations like addition, subtraction, and multiplication. These operations have specific rules:

• Matrix addition and subtraction can only occur when matrices have the same dimensions.
• Matrix multiplication requires a special condition to be met: the number of columns in the first matrix must match the number of rows in the second matrix.

Multiplying matrices is quite unlike multiplying regular numbers. Each element in the resulting matrix comes from a dot product, which involves multiplying corresponding elements and then summing them up.
Matrix algebra is especially useful for solving systems of equations, performing transformations in space, and representing complex networks in fields like computer science, economics, and engineering.

Matrix Dimensions

Matrix dimensions refer to the size of a matrix, typically expressed as "rows x columns." The order in which these dimensions are stated is crucial: rows come first, then columns.

• For example, a 3x2 matrix has 3 rows and 2 columns.
• A special square matrix has the same number of rows and columns, like the 3x3 identity matrix discussed earlier.

Understanding the dimensions is key to performing matrix operations like multiplication, which only works when the number of columns in the first matrix matches the number of rows in the second. Dimensions also help determine the resulting matrix's size, which is crucial when planning calculations in various applications like computer graphics and statistics.