Problem 28 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) BC cannot be performed; (b) BF = B.

Step 1: Determine matrix dimensions for BC

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. Matrix \(B\) is of size \(2 \times 3\) and matrix \(C\) is of size \(2 \times 3\). Since the number of columns in \(B\) (3) does not equal the number of rows in \(C\) (2), the multiplication \(BC\) cannot be performed.

Step 2: Determine matrix dimensions for BF

Matrix \(B\) is 2x3, while matrix \(F\) is a 3x3 identity matrix. Since the number of columns in \(B\) (3) equals the number of rows in \(F\) (3), the multiplication \(BF\) is possible and results in a 2x3 matrix.

Step 3: Multiply matrix B by identity matrix F

To multiply \(B\) by \(F\), note that multiplying any matrix by an identity matrix of compatible size leaves the original matrix unchanged. Therefore, the product \(BF = B\). Thus, \(BF = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix}\).

Key Concepts

Matrix Dimensions

When working with matrix multiplication, understanding matrix dimensions is crucial. Matrix dimensions are expressed as rows by columns, noted as \(m \times n\). For example, if a matrix has 2 rows and 3 columns, its dimensions are \(2 \times 3\). These dimensions determine how matrices can be multiplied.Matrix multiplication follows a specific rule: the number of columns in the first matrix must match the number of rows in the second matrix. This condition ensures that the elements of the matrices align properly for multiplication. If these dimensions do not match, the multiplication cannot proceed. For instance, matrix \(B\) has dimensions \(2 \times 3\) and matrix \(C\) also has the dimensions \(2 \times 3\). Since the columns in \(B\) (3) don't match the rows in \(C\) (2), the product \(BC\) is undefined, meaning it cannot be calculated.

Identity Matrix

The identity matrix is a special kind of matrix that acts as the multiplicative identity in matrix algebra—similar to the number 1 in basic arithmetic. An identity matrix is always square (i.e., the same number of rows and columns), and its leading diagonal (from the top left to the bottom right) is filled with 1s, while all other elements are 0. For example, a \(3 \times 3\) identity matrix \(F\) looks like this: \[\begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix}\]An important property of the identity matrix is that, when you multiply any matrix by a compatible identity matrix, the original matrix remains unchanged. In the case of matrix \(F\), multiplying \(B\) (which is \(2 \times 3\)) by \(F\) does not alter \(B\), as long as the number of columns in \(B\) equals the number of rows in \(F\). Thus, \(BF = B\), retaining the same values as the original \(B\) matrix.

Algebraic Operations

Algebraic operations in matrices include addition, subtraction, and multiplication. Each of these operations has specific rules dictated by matrix dimensions and properties.

• Addition and Subtraction: These operations require the matrices to be of the same dimensions. That means both matrices must have the same number of rows and the same number of columns. Each element in the resulting matrix is the sum or difference of the corresponding elements from the original matrices.
• Multiplication: Matrix multiplication involves taking a row from the first matrix and a column from the second matrix, multiplying their corresponding elements, and summing these products to get a single element in the resulting matrix. This operation is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.
• Scalar Multiplication: This is simpler, where every element of a matrix is multiplied by a scalar (a single number).

Matrix algebra extends these operations into more complex applications in linear transformations and other areas, making a solid understanding of these rules essential for further exploration in mathematics and related fields.