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Magazine Chapter 5: Problem 4
Let \(A=\left(\begin{array}{ll}7 & 4 \\ 2 & 1\end{array}\right)\) and \(B=\left(\begin{array}{ll}3 & 5 \\ 2 & 4\end{array}\right)\). Compute the following as effi- ciently as possible by using any of the Laws of Matrix Algebra: (a) \(A B\) (b) \(A+B\) (c) \(A^{2}+A B+B A+B^{2}\) (d) \(B^{-1} A^{-1}\) (e) \(A^{2}+A B\)
Short Answer
Expert verified
\[(a) AB = \begin{pmatrix} 29 & 51 \\ 8 & 14 \end{pmatrix}, (b) A+B = \begin{pmatrix} 10 & 9 \\ 4 & 5 \end{pmatrix}, (c) A^2 + AB + BA + B^2 = \begin{pmatrix} 136 & 135 \\ 60 & 61 \end{pmatrix}, (d) B^{-1}A^{-1} = \begin{pmatrix} -7 & 25.5 \\ 4 & -14.5 \end{pmatrix}, (e) A^2 + AB = \begin{pmatrix} 86 & 83 \\ 24 & 23 \end{pmatrix}.\]
Step by step solution
01
Compute Matrix Multiplication for AB
To find the product \( AB \), multiply each element of a row from matrix \( A \) with the corresponding element of a column from matrix \( B \), and sum them up:1. First element: \( (7 \times 3) + (4 \times 2) = 21 + 8 = 29 \)2. Second element: \( (7 \times 5) + (4 \times 4) = 35 + 16 = 51 \)3. Third element: \( (2 \times 3) + (1 \times 2) = 6 + 2 = 8 \)4. Fourth element: \( (2 \times 5) + (1 \times 4) = 10 + 4 = 14 \)Thus, \( AB = \begin{pmatrix} 29 & 51 \ 8 & 14 \end{pmatrix} \).
02
Compute Matrix Addition for A+B
Add corresponding elements of matrices \( A \) and \( B \):- First element: \( 7 + 3 = 10 \)- Second element: \( 4 + 5 = 9 \)- Third element: \( 2 + 2 = 4 \)- Fourth element: \( 1 + 4 = 5 \)Thus, \( A + B = \begin{pmatrix} 10 & 9 \ 4 & 5 \end{pmatrix} \).
03
Calculate A^2, B^2 and Combine
First, calculate \( A^2 \) and \( B^2 \).For \( A^2 = A \times A \):- First element: \( (7 \times 7) + (4 \times 2) = 49 + 8 = 57 \)- Second element: \( (7 \times 4) + (4 \times 1) = 28 + 4 = 32 \)- Third element: \( (2 \times 7) + (1 \times 2) = 14 + 2 = 16 \)- Fourth element: \( (2 \times 4) + (1 \times 1) = 8 + 1 = 9 \)\( A^2 = \begin{pmatrix} 57 & 32 \ 16 & 9 \end{pmatrix} \).For \( B^2 = B \times B \):- First element: \( (3 \times 3) + (5 \times 2) = 9 + 10 = 19 \)- Second element: \( (3 \times 5) + (5 \times 4) = 15 + 20 = 35 \)- Third element: \( (2 \times 3) + (4 \times 2) = 6 + 8 = 14 \)- Fourth element: \( (2 \times 5) + (4 \times 4) = 10 + 16 = 26 \)\( B^2 = \begin{pmatrix} 19 & 35 \ 14 & 26 \end{pmatrix} \).Now use the result from Step 1 for \( AB \) and compute \( BA \):For \( BA \):- First element: \( (3 \times 7) + (5 \times 2) = 21 + 10 = 31 \)- Second element: \( (3 \times 4) + (5 \times 1) = 12 + 5 = 17 \)- Third element: \( (2 \times 7) + (4 \times 2) = 14 + 8 = 22 \)- Fourth element: \( (2 \times 4) + (4 \times 1) = 8 + 4 = 12 \)\( BA = \begin{pmatrix} 31 & 17 \ 22 & 12 \end{pmatrix} \).Combine:\( A^2 + AB + BA + B^2 = \begin{pmatrix} 57 & 32 \ 16 & 9 \end{pmatrix} + \begin{pmatrix} 29 & 51 \ 8 & 14 \end{pmatrix} + \begin{pmatrix} 31 & 17 \ 22 & 12 \end{pmatrix} + \begin{pmatrix} 19 & 35 \ 14 & 26 \end{pmatrix} = \begin{pmatrix} 136 & 135 \ 60 & 61 \end{pmatrix} \).
04
Compute Inverses B^{-1}, A^{-1} and Find B^{-1}A^{-1}
Calculate the inverse of a 2x2 matrix \( M = \begin{pmatrix} a & b \ c & d \end{pmatrix} \):\( M^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \).For \( B \):\( ext{det}(B) = (3)(4) - (5)(2) = 12 - 10 = 2 \).\( B^{-1} = \frac{1}{2} \begin{pmatrix} 4 & -5 \ -2 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -2.5 \ -1 & 1.5 \end{pmatrix} \).For \( A \):\( ext{det}(A) = (7)(1) - (4)(2) = 7 - 8 = -1 \).\( A^{-1} = -1 \begin{pmatrix} 1 & -4 \ -2 & 7 \end{pmatrix} = \begin{pmatrix} -1 & 4 \ 2 & -7 \end{pmatrix} \).Now, compute \( B^{-1}A^{-1} \):First element: \( (2 \times -1) + (-2.5 \times 2) = -2 - 5 = -7 \).Second element: \( (2 \times 4) + (-2.5 \times -7) = 8 + 17.5 = 25.5 \).Third element: \( (-1 \times -1) + (1.5 \times 2) = 1 + 3 = 4 \).Fourth element: \( (-1 \times 4) + (1.5 \times -7) = -4 - 10.5 = -14.5 \).Thus, \( B^{-1}A^{-1} = \begin{pmatrix} -7 & 25.5 \ 4 & -14.5 \end{pmatrix} \).
05
Simplified Calculation for A^2 + AB
Use previously computed \( A^2 \) and \( AB \).\( A^2 + AB = \begin{pmatrix} 57 & 32 \ 16 & 9 \end{pmatrix} + \begin{pmatrix} 29 & 51 \ 8 & 14 \end{pmatrix} = \begin{pmatrix} 86 & 83 \ 24 & 23 \end{pmatrix} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Multiplication
Matrix multiplication is an essential concept in matrix algebra. It's like combining two sets of rules to create a new set. The process involves using each row of the first matrix and every column of the second matrix.
To perform this multiplication, you follow three main steps:
For instance, if you multiply matrices A and B, and both are 2x2, each element of the resulting matrix AB is calculated by this method. So, for matrix A's first row and matrix B's first column, you have:
\[ (A_{11} \cdot B_{11}) + (A_{12} \cdot B_{21}) \]
This method ensures you build a new matrix that retains the dimensions and key information from both original matrices.
To perform this multiplication, you follow three main steps:
- Take a row from the first matrix and a column from the second matrix.
- Multiply corresponding elements from the row and column.
- Add all those products together to get a single number in the resulting matrix.
For instance, if you multiply matrices A and B, and both are 2x2, each element of the resulting matrix AB is calculated by this method. So, for matrix A's first row and matrix B's first column, you have:
\[ (A_{11} \cdot B_{11}) + (A_{12} \cdot B_{21}) \]
This method ensures you build a new matrix that retains the dimensions and key information from both original matrices.
Matrix Addition
Matrix addition is one of the simpler operations in matrix algebra. It involves two matrices where you add corresponding elements and create a new matrix. It's important that both matrices have the same dimensions.
In the case of 2x2 matrices, like matrices A and B in our exercise, matrix addition works like this:
Let's walk through this process with matrices A and B:
This operation is not commutative. While the addition order doesn鈥檛 matter (e.g., \( A + B = B + A \)), in other operations, like multiplication, B multiplying A does not necessarily equal A multiplying B.
In the case of 2x2 matrices, like matrices A and B in our exercise, matrix addition works like this:
- Identify corresponding elements in both matrices.
- Add these corresponding elements together.
- The sum becomes the element in the resulting matrix.
Let's walk through this process with matrices A and B:
- First element: First row, first column of both matrices, resulting in \( A_{11} + B_{11} \).
- Second element: First row, second column, or \( A_{12} + B_{12} \).
This operation is not commutative. While the addition order doesn鈥檛 matter (e.g., \( A + B = B + A \)), in other operations, like multiplication, B multiplying A does not necessarily equal A multiplying B.
Matrix Inversion
Matrix inversion helps to "undo" the effect of a matrix. It's akin to finding a reciprocal in arithmetic. However, not all matrices have inverses. Only square matrices where the determinant isn鈥檛 zero can be inverted. For a 2x2 matrix, finding an inverse involves a few steps:
For matrix M = \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the inverse M-1 is:
\[ \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]
Here are the steps:
Using this approach, you can invert a matrix, enabling you to solve matrix equations or find solutions in systems represented by those matrices.
For matrix M = \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the inverse M-1 is:
\[ \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]
Here are the steps:
- Calculate the determinant, \( ad - bc \).
- If the determinant isn't zero, proceed. Otherwise, the matrix doesn鈥檛 have an inverse.
- Swap the positions of a and d, then change the signs of b and c.
- Divide all elements by the determinant.
Using this approach, you can invert a matrix, enabling you to solve matrix equations or find solutions in systems represented by those matrices.
2x2 Matrices
A 2x2 matrix is the smallest form of a square matrix, and it's especially useful in various calculations. Imagine it as a table with two rows and two columns, often used as the building blocks in matrix algebra.
Key characteristics of 2x2 matrices include:
Working with 2x2 matrices can be a pivotal step in mastering matrix operations, preparing you for more complicated forms of matrices. They establish groundwork for understanding larger matrices and complex matrix operations.
Key characteristics of 2x2 matrices include:
- They are easy to handle for both calculations and understanding basic operations.
- Their determinants can be calculated simply, as \( ad - bc \).
- These matrices frequently appear in physics, economics, and computer graphics due to their simplicity.
Working with 2x2 matrices can be a pivotal step in mastering matrix operations, preparing you for more complicated forms of matrices. They establish groundwork for understanding larger matrices and complex matrix operations.
