Problem 34
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. $$D B+D C$$
Step-by-Step Solution
Verified Answer
The result of the operation \(DB + DC\) is \(\begin{bmatrix} 38 & -15 & 35 \end{bmatrix}\).
1Step 1: Verify Matrix Dimensions for Multiplication DB
Matrix \(D\) is a \(1 \times 2\) matrix, and matrix \(B\) is a \(2 \times 3\) matrix. The number of columns in \(D\) matches the number of rows in \(B\), so their multiplication is valid, resulting in a \(1 \times 3\) matrix.
2Step 2: Perform Matrix Multiplication DB
Calculate \(DB\) by multiplying matrix \(D\) and \(B\): \[D B = \begin{bmatrix} 7 & 3 \end{bmatrix} \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} = \begin{bmatrix} (7 \cdot 3 + 3 \cdot 1) & (7 \cdot \frac{1}{2} + 3 \cdot (-1)) & (7 \cdot 5 + 3 \cdot 3) \end{bmatrix}\]\[= \begin{bmatrix} 24 & -\frac{1}{2} & 44 \end{bmatrix}\]
3Step 3: Verify Matrix Dimensions for Multiplication DC
Matrix \(D\) is a \(1 \times 2\) matrix, and matrix \(C\) is a \(2 \times 3\) matrix. The number of columns in \(D\) matches the number of rows in \(C\), so their multiplication is valid, resulting in a \(1 \times 3\) matrix.
4Step 4: Perform Matrix Multiplication DC
Calculate \(DC\) by multiplying matrix \(D\) and \(C\): \[D C = \begin{bmatrix} 7 & 3 \end{bmatrix} \begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix} = \begin{bmatrix} (7 \cdot 2 + 3 \cdot 0) & (7 \cdot -\frac{5}{2} + 3 \cdot 2) & (7 \cdot 0 + 3 \cdot -3) \end{bmatrix}\]\[= \begin{bmatrix} 14 & -\frac{29}{2} & -9 \end{bmatrix}\]
5Step 5: Verify Matrix Dimensions for Addition
Results of \(DB\) and \(DC\) are both \(1 \times 3\) matrices. Since both matrices have the same dimensions, we can perform addition.
6Step 6: Perform Matrix Addition DB + DC
Add matrices \(DB\) and \(DC\): \[D B + D C = \begin{bmatrix} 24 & -\frac{1}{2} & 44 \end{bmatrix} + \begin{bmatrix} 14 & -\frac{29}{2} & -9 \end{bmatrix} = \begin{bmatrix} (24 + 14) & (-\frac{1}{2} - \frac{29}{2}) & (44 - 9) \end{bmatrix}\]\[= \begin{bmatrix} 38 & -15 & 35 \end{bmatrix}\]
Key Concepts
Matrix MultiplicationMatrix AdditionMatrix Dimensions
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices together to produce a third matrix. For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix. If matrix \( A \) is sized \( m \times n \) and matrix \( B \) is \( n \times p \), then their product \( AB \) will be an \( m \times p \) matrix.
When multiplying matrices, each element in the resulting matrix is obtained by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. For instance, if \( D \) is a \( 1 \times 2 \) matrix and \( B \) is a \( 2 \times 3 \) matrix, as we multiply \( D \) by \( B \), we are performing a series of operations where each element of the row from \( D \) is multiplied by each corresponding element of each column in \( B \), and the results are summed to produce the elements of the resulting matrix.
Visualizing matrix multiplication might remind you of aligning rows and columns for addition and then performing multiplication like calculating individual sections. Keep in mind that matrix multiplication is not commutative. This means \( AB \) doesn’t necessarily equal \( BA \). Understanding these basic rules is crucial for correctly performing matrix multiplication.
When multiplying matrices, each element in the resulting matrix is obtained by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. For instance, if \( D \) is a \( 1 \times 2 \) matrix and \( B \) is a \( 2 \times 3 \) matrix, as we multiply \( D \) by \( B \), we are performing a series of operations where each element of the row from \( D \) is multiplied by each corresponding element of each column in \( B \), and the results are summed to produce the elements of the resulting matrix.
Visualizing matrix multiplication might remind you of aligning rows and columns for addition and then performing multiplication like calculating individual sections. Keep in mind that matrix multiplication is not commutative. This means \( AB \) doesn’t necessarily equal \( BA \). Understanding these basic rules is crucial for correctly performing matrix multiplication.
Matrix Addition
Matrix addition is a simpler operation compared to multiplication. In matrix addition, you add corresponding elements from the two matrices to form a new matrix. For two matrices \( A \) and \( B \) to be added together, they must have identical dimensions. This means the matrices must have the same number of rows and columns.
If both matrices are of size \( m \times n \), you add the elements in each position together to form a new \( m \times n \) matrix, where each element is the sum of the corresponding elements in \( A \) and \( B \).
For example, if matrix \( DB \) is \( 1 \times 3 \) and matrix \( DC \) is also \( 1 \times 3 \), these matrices can be added to create another \( 1 \times 3 \) matrix. The process is straightforward, simply add the elements: first element of \( DB \) with the first element of \( DC \), the second with the second, and so on.
Matrix addition is commutative, unlike multiplication. This means \( A + B = B + A \). It is also associative, so \( (A + B) + C = A + (B + C) \). Understanding these properties can help simplify solving matrix equations.
If both matrices are of size \( m \times n \), you add the elements in each position together to form a new \( m \times n \) matrix, where each element is the sum of the corresponding elements in \( A \) and \( B \).
For example, if matrix \( DB \) is \( 1 \times 3 \) and matrix \( DC \) is also \( 1 \times 3 \), these matrices can be added to create another \( 1 \times 3 \) matrix. The process is straightforward, simply add the elements: first element of \( DB \) with the first element of \( DC \), the second with the second, and so on.
Matrix addition is commutative, unlike multiplication. This means \( A + B = B + A \). It is also associative, so \( (A + B) + C = A + (B + C) \). Understanding these properties can help simplify solving matrix equations.
Matrix Dimensions
Matrix dimensions are central to understanding how to perform both multiplication and addition of matrices. The dimensions of a matrix tell you how many rows and columns it has, described in the format \( m \times n \) where \( m \) is the number of rows and \( n \) is the number of columns.
Knowing matrix dimensions is crucial in identifying whether certain operations like matrix multiplication and addition can be carried out. For multiplication, it’s important that the number of columns in the first matrix equals the number of rows in the second matrix. Whereas for addition, both matrices must be the same size in terms of rows and columns.
When you encounter matrices, always identify their dimensions first. This will guide you in performing the right operations. Incorrect dimensions can lead to incorrect operations and meaningless results. For example, trying to multiply two matrices without compatible dimensions might result in a false assumption of how multiplication operates on those matrices.
To summarize, always consider the matrix dimensions before proceeding with operations, as they dictate the feasibility of matrix arithmetic. This approach not only helps in avoiding errors but also provides a structured method to work through problems involving matrices.
Knowing matrix dimensions is crucial in identifying whether certain operations like matrix multiplication and addition can be carried out. For multiplication, it’s important that the number of columns in the first matrix equals the number of rows in the second matrix. Whereas for addition, both matrices must be the same size in terms of rows and columns.
When you encounter matrices, always identify their dimensions first. This will guide you in performing the right operations. Incorrect dimensions can lead to incorrect operations and meaningless results. For example, trying to multiply two matrices without compatible dimensions might result in a false assumption of how multiplication operates on those matrices.
To summarize, always consider the matrix dimensions before proceeding with operations, as they dictate the feasibility of matrix arithmetic. This approach not only helps in avoiding errors but also provides a structured method to work through problems involving matrices.
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