Problem 26 - 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) \(3 B+2 C\) (b) \(2 H+D\)

Step-by-Step Solution

Answer

(a) \(3B + 2C = \left[\begin{array}{rrr}13 & -\frac{7}{2} & 15 \\ 3 & 1 & 3\end{array}\right]\); (b) cannot be performed due to incompatible dimensions.

Step 1: Check Dimensions for 3B and 2C

Both matrices \( B \) and \( C \) are of size \( 2 \times 3 \) which means they can be added together. Each element of \( 3B \) and \( 2C \) will be calculated independently and then added to get a resulting matrix of the same size \( 2 \times 3 \).

Step 2: Calculate 3B

Multiply each element of \( B \) by 3:\[3B = \left[\begin{array}{rrr}3 \times 3 & \frac{1}{2} \times 3 & 5 \times 3 \ 1 \times 3 & -1 \times 3 & 3 \times 3\end{array}\right] = \left[\begin{array}{rrr}9 & \frac{3}{2} & 15 \ 3 & -3 & 9\end{array}\right]\]

Step 3: Calculate 2C

Multiply each element of \( C \) by 2:\[2C = \left[\begin{array}{rrr}2 \times 2 & -\frac{5}{2} \times 2 & 0 \times 2 \ 0 \times 2 & 2 \times 2 & -3 \times 2\end{array}\right] = \left[\begin{array}{rrr}4 & -5 & 0 \ 0 & 4 & -6\end{array}\right]\]

Step 4: Add 3B and 2C

Add corresponding elements of \( 3B \) and \( 2C \) to get the resulting matrix:\[3B + 2C = \left[\begin{array}{rrr}9 + 4 & \frac{3}{2} + (-5) & 15 + 0 \ 3 + 0 & -3 + 4 & 9 + (-6)\end{array}\right] = \left[\begin{array}{rrr}13 & -\frac{7}{2} & 15 \ 3 & 1 & 3\end{array}\right]\]

Step 5: Check Dimensions for 2H and D

Matrix \( H \) is \( 2 \times 2 \) and matrix D is \( 1 \times 2 \). These matrices cannot be added because their dimensions are different. Therefore, the operation \( 2H + D \) cannot be performed.

Key Concepts

Matrix Addition

Matrix addition is one of the simplest operations you can perform on matrices. To add two matrices, they must first have the exact same dimensions. This means they need to have the same number of rows and columns. If they do, you can proceed by adding corresponding elements of the two matrices together.

For instance, if you have matrices A and B, both of which are 2 x 3, you can find the resulting matrix by adding the element in row 1, column 1 of matrix A to the element in row 1, column 1 of matrix B, and so forth for each corresponding element of the matrices.

• Matrix A: \(\begin{bmatrix}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23}\end{bmatrix}\)
• Matrix B: \(\begin{bmatrix}b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23}\end{bmatrix}\)

The resulting matrix from adding A and B would be \(\begin{bmatrix}a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23}\end{bmatrix}\). Just make sure to double-check that the matrices you are adding have the same dimensions first! If not, you'll run into issues as you won't have a pair of elements to add together in some spots.

Matrix Multiplication

Matrix multiplication is a bit more complex than addition. In this operation, the dimensions of the matrices also play a crucial role, but the rules are slightly different. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.

For example, if you have matrix A which is 2 x 3, to multiply it by another matrix, matrix B, matrix B must have 3 rows. The resulting matrix will then have the number of rows of the first matrix and the number of columns of the second matrix.

• Matrix A has dimensions m x n
• Matrix B needs to have dimensions n x p

Resulting matrix will be m x p.
Matrix multiplication involves taking the dot product of rows and columns. This may seem tricky at first but can be solved systematically. For each element in the resulting matrix, you combine the elements of the row from the first matrix with the corresponding elements of the column of the second matrix.Remember, matrix multiplication is NOT commutative, meaning \( AB eq BA \). The order of multiplication matters!

Matrix Dimensions

Understanding matrix dimensions is the foundation of performing any matrix operation. A matrix's dimension is expressed as 'm x n', where 'm' is the number of rows and 'n' is the number of columns.

Before performing operations like addition or multiplication, checking dimensions is crucial because it dictates what can be done with the matrices involved.

• If two matrices share the same dimensions, they can be added or subtracted element-wise.
• For multiplication, verify if the number of columns in the first matrix matches the number of rows in the second.

When analyzing problems involving matrices, always start by identifying and comparing their dimensions. This will determine subsequent operations and ensures you're following the rules of matrix math accurately.