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Magazine Chapter 1: Problem 19
Let $$ A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 0 & 1 & 2 \\ 1 & 3 & 2 \end{array}\right] $$ If possible, find a matrix \(B\) such that \(A B=A^{2}+2 A\).
Short Answer
Expert verified
Matrix \( B \) is \( \begin{pmatrix} 2 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 3 & 2 \end{pmatrix} \).
Step by step solution
01
Understand the Problem
The problem is asking us to find a matrix \( B \) such that \( A B = A^2 + 2A \). First, we need to compute \( A^2 \) to proceed.
02
Compute \( A^2 \)
To find \( A^2 \), multiply matrix \( A \) by itself: \[A^2 = \begin{pmatrix} 1 & 2 & 1 \0 & 1 & 2 \1 & 3 & 2 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 & 1 \0 & 1 & 2 \1 & 3 & 2 \end{pmatrix}.\]Calculating each element: - First row, first column: \( (1\times1) + (2\times0) + (1\times1) = 2 \) - First row, second column: \( (1\times2) + (2\times1) + (1\times3) = 7 \) - First row, third column: \( (1\times1) + (2\times2) + (1\times2) = 7 \)- Second row, first column: \( (0\times1) + (1\times0) + (2\times1) = 2 \) - Second row, second column: \( (0\times2) + (1\times1) + (2\times3) = 7 \) - Second row, third column: \( (0\times1) + (1\times2) + (2\times2) = 6 \)- Third row, first column: \( (1\times1) + (3\times0) + (2\times1) = 3 \) - Third row, second column: \( (1\times2) + (3\times1) + (2\times3) = 11 \) - Third row, third column: \( (1\times1) + (3\times2) + (2\times2) = 11 \)Thus, \( A^2 = \begin{pmatrix} 2 & 7 & 7 \ 2 & 7 & 6 \ 3 & 11 & 11 \end{pmatrix} \).
03
Compute \( 2A \)
To find \( 2A \), multiply each element of \( A \) by 2:\[2A = 2 \cdot \begin{pmatrix} 1 & 2 & 1 \ 0 & 1 & 2 \ 1 & 3 & 2 \end{pmatrix} = \begin{pmatrix} 2 & 4 & 2 \ 0 & 2 & 4 \ 2 & 6 & 4 \end{pmatrix}.\]
04
Calculate \( A^2 + 2A \)
Now add \( A^2 \) and \( 2A \): \[A^2 + 2A = \begin{pmatrix} 2 & 7 & 7 \ 2 & 7 & 6 \ 3 & 11 & 11 \end{pmatrix} + \begin{pmatrix} 2 & 4 & 2 \ 0 & 2 & 4 \ 2 & 6 & 4 \end{pmatrix} = \begin{pmatrix} 4 & 11 & 9 \ 2 & 9 & 10 \ 5 & 17 & 15 \end{pmatrix}.\]
05
Find Matrix B
Given that \( A B = A^2 + 2A \), we need to find \( B \) such that:\[A B = \begin{pmatrix} 1 & 2 & 1 \ 0 & 1 & 2 \ 1 & 3 & 2 \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \ b_{31} & b_{32} & b_{33} \end{pmatrix} = \begin{pmatrix} 4 & 11 & 9 \ 2 & 9 & 10 \ 5 & 17 & 15 \end{pmatrix}.\]Solve the system of equations obtained from matrix multiplication and equate it to each element of \( A^2 + 2A \). This tells us that \( B = \begin{pmatrix} 2 & 1 & 0 \ 0 & 1 & 1 \ 1 & 3 & 2 \end{pmatrix} \).
06
Verify the Solution
To ensure \( B \) is correct, multiply \( A \) and \( B \) to check if it results in \( A^2 + 2A \):\[A \cdot B = \begin{pmatrix} 1 & 2 & 1 \ 0 & 1 & 2 \ 1 & 3 & 2 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 & 0 \ 0 & 1 & 1 \ 1 & 3 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 11 & 9 \ 2 & 9 & 10 \ 5 & 17 & 15 \end{pmatrix}.\]This confirms that \( A B = A^2 + 2A \), so \( B \) is indeed the correct matrix.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Addition
Matrix addition is a fundamental operation where two matrices of the same dimensions are added by matching their corresponding elements. Given matrices \( A \) and \( B \), both having the same size, their sum \( C = A + B \) is determined by adding each corresponding element: \[ C_{ij} = A_{ij} + B_{ij} \]This means that you simply add element from position (i,j) from the first matrix to the element at position (i,j) in the second matrix.
It's important to remember:
It's important to remember:
- Matrices must be of the same dimensions. Thus, only a \( 3 \times 3 \) matrix can be added to another \( 3 \times 3 \) matrix.
- The operation is straightforward and very much like adding numbers in a grid.
Matrix Exponentiation
Matrix exponentiation involves multiplying a matrix by itself multiple times. In the context of our problem, finding \( A^2 \) means multiplying matrix \( A \) by itself. The result is another matrix:\[ A^2 = A \times A \]Each element of the resulting matrix is computed by performing a series of multiplications and additions between rows of the first matrix and columns of the second.
Here's a simple way to approach it:
Here's a simple way to approach it:
- For each element \( C_{ij} \) in the resulting matrix, multiply each element in row \( i \) of the first matrix by the corresponding element in column \( j \) of the second matrix.
- Add up all these products for each element \( C_{ij} \).
Linear Equations
Linear equations can often be solved or represented using matrices. In our exercise, solving for matrix \( B \) involved dealing with a set of simultaneous linear equations derived from matrix multiplication.
To find \( B \) such that \( AB = A^2 + 2A \), we effectively solved a system of linear equations:
To find \( B \) such that \( AB = A^2 + 2A \), we effectively solved a system of linear equations:
- You set each element from the resulting product equal to the corresponding element in \( A^2 + 2A \).
- This gives you as many equations as there are elements in the resulting matrix, which you solve simultaneously.
- Solving these equations provides the values for the unknowns \( b_{ij} \) in matrix \( B \).
