Problem 36 - 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) \(D B+D C\) (b) \(B F+F E\)

Step-by-Step Solution

Answer

(a) DB + DC = [38, -10, 35]. (b) BF and FE cannot be added due to dimension mismatch.

Step 1: Check Compatibility for DB and DC

Matrix \(D\) is a \(1 \times 2\) matrix and \(B\) is a \(2 \times 3\) matrix. To multiply \(D\) by \(B\), the inner dimensions must match, which they do (2 from \(D\) matches 2 from \(B\)), resulting in a \(1 \times 3\) matrix. Similarly, matrix \(C\) is also a \(2 \times 3\) matrix, so \(DC\) is valid and results in a \(1 \times 3\) matrix. Therefore, \(DB + DC\) is a valid operation.

Step 2: Calculate DB

To find \(DB\), multiply each element of row in \(D\) by corresponding columns in \(B\): \[ DB = \left[\begin{array}{lll}7 & 3\end{array}\right] \left[\begin{array}{rrr}3 & \frac{1}{2} & 5 \ 1 & -1 & 3\end{array}\right] = \left[\begin{array}{lll}(7 \times 3 + 3 \times 1) & (7 \times \frac{1}{2} + 3 \times -1) & (7 \times 5 + 3 \times 3)\end{array}\right] = \left[\begin{array}{lll}24 & 2.5 & 44\end{array}\right] \]

Step 3: Calculate DC

Similarly, multiply \(D\) by \(C\):\[ DC = \left[\begin{array}{lll}7 & 3\end{array}\right]\left[\begin{array}{rrr}2 & -\frac{5}{2} & 0 \ 0 & 2 & -3\end{array}\right] = \left[\begin{array}{lll}(7 \times 2 + 3 \times 0) & (7 \times -\frac{5}{2} + 3 \times 2) & (7 \times 0 + 3 \times -3)\end{array}\right] = \left[\begin{array}{lll}14 & -\frac{23}{2} & -9\end{array}\right] \]

Step 4: Add DB and DC

Finally, add the results from the previous steps:\[DB + DC = \left[\begin{array}{lll}24 & 2.5 & 44\end{array}\right] + \left[\begin{array}{lll}14 & -\frac{23}{2} & -9\end{array}\right]= \left[\begin{array}{lll}38 & -10 & 35\end{array}\right]\]

Step 5: Check Compatibility for BF and FE

Matrix \(B\) is \(2 \times 3\), and matrix \(F\) is \(3 \times 3\), allowing the multiplication \(BF\) since the inner dimensions match. This results in a \(2 \times 3\) matrix. Matrix \(F\) is \(3 \times 3\) and \(E\) is \(3 \times 1\), allowing multiplication \(FE\) to give a \(3 \times 1\) matrix. However, since \(BF\) and \(FE\) have different dimensions, adding them is not possible.

Step 6: Evaluate BF and Explain why FE + BF isn't possible

Compute \(BF\): Since \(F\) is an identity matrix, \(BF = B\). \[ BF = \left[\begin{array}{rrr}3 & \frac{1}{2} & 5 \ 1 & -1 & 3\end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right] = \left[\begin{array}{rrr}3 & \frac{1}{2} & 5 \ 1 & -1 & 3\end{array}\right]\]Since the result of \(BF\) is a \(2 \times 3\) matrix and \(FE\) would be a \(3 \times 1\) matrix after computation, these cannot be added, as their sizes do not match.

Key Concepts

Matrix Multiplication

Matrix multiplication is a core operation that can be performed on two matrices. However, for two matrices to be multiplied, their dimensions must be compatible. Specifically, the number of columns in the first matrix must equal the number of rows in the second matrix. This property of compatibility is often referred to as the 'inner dimensions' matching rule. For example, to multiply a matrix D (which is 1x2) with matrix B (which is 2x3), the numbers of columns in D and rows in B must match, which they do in this case (2 and 2). When the dimensions align this way, the result is a new matrix with dimensions based on the 'outer dimensions'. For this multiplication, the resulting matrix is 1x3. To execute matrix multiplication, follow these steps:

• Take each row of the first matrix and each column of the second matrix.
• Multiply corresponding elements and then add them up to get a single number.
• Repeat this process for all rows and columns.

Matrix Addition

Matrix addition is a straightforward operation, but it comes with a strict requirement: both matrices involved must have the exact same dimensions. This means that you can only add matrices that have the same number of rows and columns. During the addition, each element of one matrix is added to the corresponding element of the other matrix. It is essentially a straightforward element-wise operation. For instance, if you have two matrices A and B, both sized 2x3, then the elements at position (1,1) in both matrices will be added together to produce the element at position (1,1) in the resulting matrix. It’s important to remember that without matching dimensions, the matrices cannot be added. For example, trying to add a 2x3 matrix to a 3x1 matrix is not possible.

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrices. Dimensions define the size and shape of a matrix, describing how many rows and columns it has. This information is often represented in the form of 'rows × columns', such as 2x3 for a matrix with 2 rows and 3 columns. Dimensions are significant because they affect which operations are possible. Certain operations, like multiplication or addition, require specific conditions about matrix dimensions. For multiplication, the internal dimension, or the number of columns in the first matrix, must match the number of rows in the second matrix. When adding matrices, their dimensions must match exactly, meaning both must be, for example, 3x3 or 2x2. Being aware of matrix dimensions helps to quickly determine the viability of potential operations.

Identity Matrix

The identity matrix is a special kind of matrix that has a pivotal role in matrix multiplication. It is a square matrix, meaning it has the same number of rows and columns, typically denoted by the symbol 'I'. The defining characteristic of an identity matrix is that it has 1s on the diagonal from the top-left to the bottom-right and 0s elsewhere. The identity matrix acts as the multiplicative identity in the world of matrices. When any matrix is multiplied by the identity matrix of the same dimension, the original matrix is returned unchanged. This property resembles how multiplying a number by 1 leaves the number unchanged in regular arithmetic. For example, multiplying a 3x3 matrix by a 3x3 identity matrix results in the original matrix itself. Knowing how an identity matrix functions is essential for simplifying matrix operations and maintaining the integrity of the operands involved in complex calculations.