Math Problem Statement
Find the following matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations.
(a)
Start 2 By 2 Table 1st Row 1st Column 1 2nd Column 0 2nd Row 1st Column 0 2nd Column 1 EndTable Start 2 By 2 Table 1st Row 1st Column negative 4 2nd Column 3 2nd Row 1st Column negative 2 2nd Column negative 5 EndTable
1
0
0
1
−4
3
−2
−5
(b) Start 2 By 2 Table 1st Row 1st Column negative 4 2nd Column 3 2nd Row 1st Column negative 2 2nd Column negative 5 EndTable Start 2 By 2 Table 1st Row 1st Column 1 2nd Column 0 2nd Row 1st Column 0 2nd Column 1 EndTable
−4
3
−2
−5
1
0
0
1
Question content area bottom
Part 1
(a)
Start 2 By 2 Table 1st Row 1st Column 1 2nd Column 0 2nd Row 1st Column 0 2nd Column 1 EndTable Start 2 By 2 Table 1st Row 1st Column negative 4 2nd Column 3 2nd Row 1st Column negative 2 2nd Column negative 5 EndTable
1
0
0
1
−4
3
−2
−5
equals=enter your response here
Part 2
(b)
Start 2 By 2 Table 1st Row 1st Column negative 4 2nd Column 3 2nd Row 1st Column negative 2 2nd Column negative 5 EndTable Start 2 By 2 Table 1st Row 1st Column 1 2nd Column 0 2nd Row 1st Column 0 2nd Column 1 EndTable
−4
3
−2
−5
1
0
0
1
equals=enter your response here
Solution
Let's solve the given matrix products:
Part (a)
We are asked to find the product of the following two matrices:
1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix}$$ The first matrix is the identity matrix. When you multiply any matrix by the identity matrix, the result is the original matrix itself. Therefore: $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix} = \begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix}$$ So the answer for part (a) is: $$\begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix}$$ ### Part (b) Now we need to find the product of the following two matrices: $$\begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ Again, the second matrix is the identity matrix. Multiplying any matrix by the identity matrix results in the original matrix itself. Therefore: $$\begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix}$$ So the answer for part (b) is: $$\begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix}$$ ### Summary Both products result in the matrix: $$\begin{pmatrix} -4 & 3 \\ -2 & -5 \end{pmatrix}$$ Do you have any questions or need further details? Here are five related questions you might find interesting: 1. What is the result of multiplying any matrix by a zero matrix? 2. How does the multiplication of matrices differ from the multiplication of numbers? 3. How do we find the inverse of a 2x2 matrix? 4. What properties does the identity matrix have? 5. Can matrix multiplication be commutative under certain conditions? **Tip:** Remember that the identity matrix acts as the "1" in matrix multiplication, leaving any matrix it multiplies unchanged.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Identity Matrix
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12