Problem 30 - 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) \(B C\) (b) \(B F\)

Step-by-Step Solution

Answer

(a) BC is not possible. (b) BF = B.

Step 1: Determine the Size of Matrix B

Matrix \( B \) is a matrix with 2 rows and 3 columns, so it is a \(2 \times 3\) matrix.

Step 2: Determine the Size of Matrix C

Matrix \( C \) is also a matrix with 2 rows and 3 columns, so it is a \(2 \times 3\) matrix.

Step 3: Check Compatibility for Matrix Multiplication BC

For the product \( BC \) to be defined, the number of columns in \( B \) must equal the number of rows in \( C \). However, \( B \) has 3 columns and \( C \) has 2 rows, hence the multiplication is not possible.

Step 4: Determine the Size of Matrix F

Matrix \( F \) is a \(3 \times 3\) identity matrix, meaning it has 3 rows and 3 columns.

Step 5: Check Compatibility for Matrix Multiplication BF

For the product \( BF \) to be defined, the number of columns in \( B \) must equal the number of rows in \( F \). Since \( B \) is \(2 \times 3\) and \( F \) is \(3 \times 3\), the product \( BF \) is possible.

Step 6: Multiply B by F

Since \( F \) is the identity matrix, the operation \( BF \) simplifies to matrix \( B \) itself. The result is \[BF = \left[\begin{array}{ccc}3 & \frac{1}{2} & 5 \1 & -1 & 3\end{array}\right]\]

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial in matrix operations like multiplication. Each matrix is essentially a grid of numbers organized into rows and columns. When we describe a matrix, we use the notation \( m \times n \), where \( m \) represents the number of rows and \( n \) signifies the number of columns. For instance, if matrix \( B \) is a \( 2 \times 3 \) matrix, it means it has 2 rows and 3 columns.
Matrix dimensions play a significant role in determining whether operations such as multiplication are possible. This is because matrix multiplication is not as straightforward as number or vector multiplication; specific rules based on dimensions must be followed. These dimensions dictate how we align the values across the matrices for proper computation.

Identity Matrix

An identity matrix is a special type of matrix that is the equivalent of the number 1 in matrix arithmetic. When any matrix is multiplied by an identity matrix, it remains unchanged. The identity matrix is denoted as \( I \) and is square—meaning it has the same number of rows and columns. For example, a 3x3 identity matrix \( F \) looks like this:

• All the entries on its main diagonal (from upper left to lower right) are 1.
• All other entries are 0.

Thus, when matrix \( B \) is multiplied by matrix \( F \), the result is matrix \( B \) itself. This is because the identity matrix does not alter or add any extra components to the multiplication process, much like multiplying a number by 1.

Matrix Compatibility

Matrix compatibility is key to performing matrix operations such as multiplication. To multiply two matrices, you need to ensure their dimensions align in a specific way: the number of columns in the first matrix must match the number of rows in the second matrix. This prerequisite can often determine whether or not a multiplication can proceed.
For example, if matrix \( B \) is \( 2 \times 3 \) and matrix \( C \) is also \( 2 \times 3 \), you cannot multiply \( B \) by \( C \) because the number of columns in \( B \) (which is 3) does not match the number of rows in \( C \) (which is 2). Conversely, multiplication between matrix \( B \) (a \( 2 \times 3 \) matrix) and matrix \( F \) (a \( 3 \times 3 \) matrix) is indeed possible because the columns of \( B \) match the rows of \( F \), leading to a meaningful computation.