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Chapter 2: Problem 10

Let \(A=\left[\begin{array}{rr}{2} & {-3} \\ {-4} & {6}\end{array}\right], B=\left[\begin{array}{cc}{8} & {4} \\ {5} & {5}\end{array}\right],\) and \(C=\left[\begin{array}{rr}{5} & {-2} \\ {3} & {1}\end{array}\right]\) Verify that \(A B=A C\) and yet \(B \neq C\)

Short Answer

Expert verified
Yes, \(AB = AC\) and \(B \neq C\).

Step by step solution

01

Matrix Multiplication of A and B

Calculate the product of matrices \(A\) and \(B\). The formula for matrix multiplication is: \( (AB)_{ij} = \sum_{k} a_{ik}b_{kj} \). Thus, we compute: \[ AB = \left[\begin{array}{cc}2 & -3 \ -4 & 6\end{array}\right] \left[\begin{array}{cc}8 & 4 \ 5 & 5\end{array}\right] = \left[ \begin{array}{cc} 2\cdot8 + (-3)\cdot5 & 2\cdot4 + (-3)\cdot5 \ -4\cdot8 + 6\cdot5 & -4\cdot4 + 6\cdot5 \end{array}\right] = \left[ \begin{array}{cc} 16 - 15 & 8 -15 \ -32 + 30 & -16 + 30 \end{array} \right] = \left[\begin{array}{cc}1 & -7 \ -2 & 14\end{array}\right] \]
02

Matrix Multiplication of A and C

Next, we need to calculate the product of matrices \(A\) and \(C\). Again, use matrix multiplication: \[ AC = \left[\begin{array}{cc}2 & -3 \ -4 & 6\end{array}\right] \left[\begin{array}{cc}5 & -2 \ 3 & 1\end{array}\right] = \left[ \begin{array}{cc} 2\cdot5 + (-3)\cdot3 & 2\cdot(-2) + (-3)\cdot1 \ -4\cdot5 + 6\cdot3 & -4\cdot(-2) + 6\cdot1 \end{array} \right] = \left[ \begin{array}{cc} 10 - 9 & -4 - 3 \ -20 + 18 & 8 + 6 \end{array}\right] = \left[\begin{array}{cc} 1 & -7 \ -2 & 14\end{array}\right] \]
03

Compare AB and AC Results

Now, compare the results obtained from matrices \(AB\) and \(AC\). Since both are \(\left[\begin{array}{cc} 1 & -7 \ -2 & 14\end{array}\right]\), it verifies that \(AB = AC\).
04

Verify B is Not Equal to C

To complete the verification, compare matrices \(B\) and \(C\):\[ B = \left[\begin{array}{cc}8 & 4 \ 5 & 5\end{array}\right], \quad C = \left[\begin{array}{cc}5 & -2 \ 3 & 1\end{array}\right] \]Since the corresponding elements of \(B\) and \(C\) are not equal (e.g., \(8 eq 5\)), it confirms that \(B eq C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors and matrices, among other concepts. Matrices are rectangular arrays of numbers, and they can be used to represent and solve linear equations. This area of math is crucial for many fields, including engineering, physics, computer science, and more.
Linear algebra provides tools for manipulating these matrices through operations such as addition, subtraction, and multiplication. Understanding these operations is key because they help define the relationships between different mathematical entities.
In the exercise provided, matrix multiplication is used to test whether true equality holds between the products of different matrices and to further understand their properties. This highlights the importance of linear algebra in problem-solving and the verification of matrix equations.
Matrix Equality
Matrix equality is an important concept in linear algebra. Two matrices are considered equal if they have the same dimensions and each corresponding element is identical. This is crucial because matrix equality helps determine whether two sets of transformations or operations yield the same result.
The exercise had us compare the result of multiplying Matrix A with Matrix B and Matrix A with Matrix C. Both multiplications resulted in the same matrix, demonstrating equivalence in their outputs. However, this doesn't imply the original matrices B and C are equal. Equality only holds in multiplication results, not necessarily in the individual matrices.
When comparing matrices, always check each element at corresponding positions. If any elements differ, the matrices are not equal. Understanding matrix equality can help you verify solutions and is essential when working with mathematical models that involve matrices.
Matrix Properties
Matrices have various properties that make them intriguing and powerful for mathematical expressions and computations.
One key property relevant to this exercise is the distributive property, which allows matrix multiplication and addition to be comparable to that of real numbers. When multiplying matrices, the order of operations must be considered since it can impact the outcome. This associative property is crucial in matrix multiplication because you cannot change the order of multiplication as you can with numbers.
Transpose, inverse, and identity are other properties to explore, giving matrices their versatile applications in linear transformations, systems of equations, and other scientific models.
In the exercise, although matrices B and C were shown not to be equal, their multiplication results when paired with the same matrix A yielded the same product. This illustrates how the associative property of matrix multiplication can be both a tool and a consideration in problem-solving analysis.

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Most popular questions from this chapter

In Exercises 11 and \(12,\) give integers \(p\) and \(q\) such that Nul \(A\) is a subspace of \(\mathbb{R}^{p}\) and \(\mathrm{Col} A\) is a subspace of \(\mathbb{R}^{q}\) . $$ A=\left[\begin{array}{rrr}{1} & {2} & {3} \\ {4} & {5} & {7} \\ {-5} & {-1} & {0} \\ {2} & {7} & {11}\end{array}\right] $$

Suppose \(T\) and \(U\) are linear transformations from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\) such that \(T(U \mathbf{x})=\mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n} .\) Is it true that \(U(T \mathbf{x})=\mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n}\) ? Why or why not?

A rotation in \(\mathbb{R}^{2}\) usually requires four multiplications. Compute the product below, and show that the matrix for a rotation can be factored into three shear transformations (each of which requires only one multiplication). $$ \left[\begin{array}{ccc}{1} & {-\tan \varphi / 2} & {0} \\ {0} & {1} & {0} \\\ {0} & {0} & {1}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {0} \\\ {\sin \varphi} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ \left[\begin{array}{ccc}{1} & {-\tan \varphi / 2} & {0} \\ {0} & {1} & {0} \\\ {0} & {0} & {1}\end{array}\right] $$

Let \(A=L U\) be an LU factorization. Explain why \(A\) can be row reduced to \(U\) using only replacement operations. (This fact is the converse of what was proved in the text.)

IM Suppose an experiment leads to the following system of equations: \(4.5 x_{1}+3.1 x_{2}=19.249\) 1.6 \(x_{1}+1.1 x_{2}=6.843\) a. Solve system \((3),\) and then solve system \((4),\) below, in which the data on the right have been rounded to two decimal places. In each case, find the exact solution. \(4.5 x_{1}+3.1 x_{2}=19.25\) 1.6 \(x_{1}+1.1 x_{2}=6.84\) b. The entries in \((4)\) differ from those in \((3)\) by less than .05\(\% .\) Find the percentage error when using the solution of \((4)\) as an approximation for the solution of \((3) .\) c. Use your matrix program to produce the condition number of the coefficient matrix in \((3) .\)
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