Problem 37
Question
If a matrix \(\mathbf{A}\) is premultiplied by an elementary matrix \(\mathbf{E}\), the product EA will be that matrix obtained from A by performing the elementary row operation symbolized by \(\mathbf{E}\). Compute the given product for an arbitrary \(3 \times 3\) \(\operatorname{matrix} \mathbf{A}\). \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & c & 1\end{array}\right) \mathbf{A}\)
Step-by-Step Solution
Verified Answer
\( \mathbf{E A} \) is \( \mathbf{A} \) with \( c \) times the second row added to the third row.
1Step 1: Identify the Elementary Matrix
The elementary matrix provided is \( \mathbf{E} = \left(\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & c & 1\end{array}\right) \). This matrix represents the row operation of adding \( c \) times the second row of \( \mathbf{A} \) to its third row.
2Step 2: Define an Arbitrary 3x3 Matrix \( \mathbf{A} \)
Let's consider the matrix \( \mathbf{A} = \left(\begin{array}{ccc}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right) \). This is a generic \( 3 \times 3 \) matrix with elements \( a_{ij} \), where \( i \) and \( j \) run from 1 to 3.
3Step 3: Perform the Matrix Multiplication \( \mathbf{E} \mathbf{A} \)
Multiply \( \mathbf{E} \) by \( \mathbf{A} \):- First row of \( \mathbf{E A} \): Multiply the first row of \( \mathbf{E} \) by each column of \( \mathbf{A} \), which results in \( (a_{11}, a_{12}, a_{13}) \).- Second row of \( \mathbf{E A} \): Multiply the second row of \( \mathbf{E} \) by each column of \( \mathbf{A} \), which results in \( (a_{21}, a_{22}, a_{23}) \).- Third row of \( \mathbf{E A} \): Multiply the third row of \( \mathbf{E} \) by each column of \( \mathbf{A} \), which results in \( (a_{31} + c \cdot a_{21}, a_{32} + c \cdot a_{22}, a_{33} + c \cdot a_{23}) \).Therefore, the resulting matrix \( \mathbf{E} \mathbf{A} \) is given by:\[ \mathbf{E} \mathbf{A} = \left(\begin{array}{ccc} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} + c \cdot a_{21} & a_{32} + c \cdot a_{22} & a_{33} + c \cdot a_{23} \end{array}\right) \]
4Step 4: Interpret the Result
The product matrix \( \mathbf{E A} \) is the original matrix \( \mathbf{A} \) with the third row modified. Specifically, \( c \) times the second row has been added to the third row.
Key Concepts
Matrix MultiplicationRow Operations3x3 Matrix
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a new matrix. This operation is not as straightforward as simple arithmetic multiplication.
It requires a specific set of rules, which make it both powerful and essential in mathematical computations.
Each element of the product matrix is calculated by taking the dot product of the corresponding row of the first matrix and the column of the second matrix. For example, if we have two matrices: \( extbf{E} \) and \( extbf{A} \), the element in the first row, first column of the resulting matrix \( extbf{EA} \) will be found by multiplying the relevant components of the first row in \( extbf{E} \) by the appropriate components of the first column in \( extbf{A} \) and adding them up.
It requires a specific set of rules, which make it both powerful and essential in mathematical computations.
Each element of the product matrix is calculated by taking the dot product of the corresponding row of the first matrix and the column of the second matrix. For example, if we have two matrices: \( extbf{E} \) and \( extbf{A} \), the element in the first row, first column of the resulting matrix \( extbf{EA} \) will be found by multiplying the relevant components of the first row in \( extbf{E} \) by the appropriate components of the first column in \( extbf{A} \) and adding them up.
- In matrix multiplication, it’s important to remember that it is not generally commutative; i.e., \( extbf{E} \textbf{A} eq \textbf{A} \textbf{E} \).
- The number of columns in the first matrix must equal the number of rows in the second matrix.
Row Operations
Row operations are maneuvers that can be applied to the rows of a matrix. They are used to simplify problems in linear algebra, such as gaussian elimination, which makes solving systems of equations more manageable.
These operations include swapping rows, scaling rows by a nonzero number, and adding or subtracting the multiples of rows.
By converting them into systematic procedures, these operations facilitate effective computation of solutions.
In the context of elementary matrices, row operations translate into specific actions:
For the problem at hand, we added \( c \) times the second row of \( \textbf{A} \) to the third row using the elementary matrix \( \textbf{E} \). This changes the third row while keeping the rest of the matrix intact.
These operations include swapping rows, scaling rows by a nonzero number, and adding or subtracting the multiples of rows.
By converting them into systematic procedures, these operations facilitate effective computation of solutions.
In the context of elementary matrices, row operations translate into specific actions:
- Row swapping: An elementary matrix that swaps two rows moves them as per the required positions.
- Row scaling: An operation where a row is multiplied by a scalar, altering its values by that multiple.
- Row addition/subtraction: This involves adding a multiple of one row to another row, as seen when we use matrices like \( extbf{E} = \left(\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & c & 1\end{array}\right) \).
For the problem at hand, we added \( c \) times the second row of \( \textbf{A} \) to the third row using the elementary matrix \( \textbf{E} \). This changes the third row while keeping the rest of the matrix intact.
3x3 Matrix
A 3x3 matrix is simply a matrix consisting of three rows and three columns. It represents a system with three equations each involving three variables.
These matrices are prominent in many mathematical applications and appear commonly in exercises due to their computational manageability relative to larger matrices.
The structure of a 3x3 matrix is usually indicated as follows:\[\begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}\]
In exercises involving row operations and matrix multiplication, especially with elementary matrices, a 3x3 matrix provides a manageable size to demonstrate these transformations without overwhelming complexity.
Thus, students can gain a clearer understanding of how elementary operations affect matrices in a tangible way.
These matrices are prominent in many mathematical applications and appear commonly in exercises due to their computational manageability relative to larger matrices.
The structure of a 3x3 matrix is usually indicated as follows:\[\begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}\]
- Each element \( a_{ij} \) corresponds to the intersection of the i-th row and j-th column.
- This form is used to denote linear transformations in three-dimensional space, among other uses.
In exercises involving row operations and matrix multiplication, especially with elementary matrices, a 3x3 matrix provides a manageable size to demonstrate these transformations without overwhelming complexity.
Thus, students can gain a clearer understanding of how elementary operations affect matrices in a tangible way.
Other exercises in this chapter
Problem 37
Show that if \(\mathbf{A}\) is a nonsingular matrix, then \(\operatorname{det} \mathbf{A}^{-1}=1 /\) det \(\mathbf{A}\).
View solution Problem 37
Let \(\mathbf{A}=\left(\begin{array}{rr}3 & -4 \\ 1 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}7 & 4 \\ -1 & -5\end{array}\right)\). Verify
View solution Problem 37
Use Cramer's rule to solve the system $$ \begin{aligned} &X=x \cos \theta+y \sin \theta \\ &Y=-x \sin \theta+y \cos \theta \end{aligned} $$ for \(x\) and \(y\).
View solution Problem 37
$$ \text { Show that if } \mathbf{A} \text { is a nonsingular matrix, then } \operatorname{det} \mathbf{A}^{-1}=1 / \operatorname{det} \mathbf{A} \text {. } $$
View solution