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Chapter 9: Problem 24

Find \(\boldsymbol{A}^{-1}\) by forming \([\boldsymbol{A} | \boldsymbol{I}]\) and then using row operations to obtain \([I | B],\) where \(A^{-1}=[B] .\) Check that \(A A^{-1}=I\) and \(A^{-1} A=I\) $$ \text { 24. } A=\left[\begin{array}{rrr} {2} & {4} & {-4} \\ {1} & {3} & {-4} \\ {2} & {4} & {-3} \end{array}\right] $$

Short Answer

Expert verified
The inverse of a provided matrix is obtained after performing relevant row operations on an augmented matrix [A | I] so as to achieve the [I | B] form. After the extraction of matrix B, it is verified that B is the inverse of A by confirming that \(AB = I\) and \(BA = I\).

Step by step solution

01

Form Augmented Matrix

Create an augmented matrix by combining the given matrix \(A\) and the identity matrix \(I\) of same order.
02

Transform into Row Echelon Form

Perform row operations so that the left side of the augmented matrix resembles an identity matrix. Typically, the row operations are: Swap two rows, Multiply a row by a non-zero scalar, Add a multiple of one row to another row. Keep doing these operations until the left side of the augmented matrix transforms into an identity matrix.
03

Extract Inverse Matrix

After successfully obtaining the [I | B] form, extract the matrix \(B\), i.e., the right side of the augmented matrix which represents the inverse of the given matrix \(A\).
04

Verification

Multiply the given matrix \(A\) with the extracted inverse \(B = A^{-1}\) from both sides and ensure that you get the identity matrix in both cases, i.e., \(AB = I\) and \(BA = I\).

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Key Concepts

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

Row Operations
Row operations are fundamental techniques used in linear algebra to manipulate matrices. They play a critical role when attempting to invert a matrix. In matrix inversion, row operations help transform the given matrix into a form where it can easily reveal its inverse.
  • Swapping Rows: You may interchange any two rows to move elements into more advantageous positions, often to get zeros below a pivot for upper triangular form.

  • Row Multiplication: A row can be multiplied by a non-zero scalar, which can simplify matrix elements and help establish leading ones in a row.

  • Adding Rows: By adding or subtracting multiples of rows, you can create zeros in specific positions, helping to reach the desired echelon form.
Understanding these operations is crucial because they maintain the underlying properties of the matrix while reshaping it into a form that reveals its inverse.
Augmented Matrix
An augmented matrix combines two matrices into one, which is extremely useful in solving systems of linear equations and finding matrix inverses. In the context of matrix inversion, this involves combining the original matrix, denoted as \(A\), with the identity matrix of the same size.
  • The matrix \(A\) represents the coefficients from a system of equations, or the matrix you wish to invert.

  • The identity matrix, \(I\), represents the baseline for the solution, as it helps extract the inverse on completion.
By carefully manipulating this augmented form using row operations, you can systematically convert the \([A | I]\) augmented matrix to the impressive \([I | B]\) form, where \(B\) is the sought-after inverse of \(A\).
The process relies on converting the left block of the augmented matrix into the identity matrix while transforming the right block into the inverse.
Identity Matrix
The identity matrix is a special kind of square matrix that plays a pivotal role in linear algebra, especially in matrix inversion. It acts as the "one" in matrix multiplication, similar to how the number one acts in numerical multiplication.
  • The identity matrix, typically denoted as \(I\), has ones on the diagonal and zeros elsewhere. For a matrix of size \(n\times n\), it is represented as: \[ I = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix} \]

  • Its crucial property is that when any matrix \(A\) of the same dimensions is multiplied by \(I\), it remains unchanged: \(AI = A\) and \(IA = A\).
Inverting a matrix essentially involves transforming it into an identity matrix using the augmented matrix \([A | I]\) and applying row operations. The identity matrix's structure allows the isolation of the inverse in the augmented system, demonstrating how \(A^{-1}\) reverses the effects of \(A\) itself.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra and plays a key role in verifying the correctness of an inverse matrix. It involves combining two matrices to produce a new matrix by systematically computing the dot product of rows from the first with columns from the second.
  • For matrices \(A\) and \(B\), the product \(AB\) results in a new matrix, contingent on the compatibility of dimensions (e.g., if \(A\) is \(m\times n\), then \(B\) should be \(n\times p\)).

  • The element in the \(i^{th}\) row and \(j^{th}\) column of the resultant matrix is calculated by taking the sum of the products of corresponding elements: \( (AB)_{ij} = \sum_{k} A_{ik}B_{kj} \).
This operation is pivotal in verifying whether a matrix inversion is correct. After identifying \(B = A^{-1}\), if \(AB = I\) and \(BA = I\), where \(I\) is the identity matrix, the inverse computation is confirmed correct.
This operation checks and balances the accuracy of inversions, grounding them in principles of matrix algebra.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using row operations on an augmented matrix, I obtain a row in which 0 s appear to the left of the vertical bar, but 6 appears on the right, so the system I'm working with has no solution.

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. a. If \(A=\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right],\) find \(A B\). b. Graph the object represented by matrix \(A B .\) What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(B ?\)

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. A furniture company produces three types of desks: a children's model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table. $$ \begin{array}{ccc} {} & {\text { Children's }} & {\text { Office }} & {\text { Deluxe }} \\\& {\text { Model }} & {\text { Model }} & {\text { Model }} \\ {\text { Cutting }} & {2 \text { hr }} & {3 \text { hr }} & {2 \text { hr }} \\\ {\text { Construction }} & {2 \text { hr }} & {1 \text { hr }} & {3 \text { hr }} \\ {\text { Finishing }} & {1 \text { hr }} & {1 \text { hr }} & {2 \text { hr }} \end{array} $$ Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

Solve and graph the solution set on a number line: $$ |2 x+3| \leq 13 $$ (Section 1.7, Example 8)

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{ll} {2} & {4} \\ {3} & {1} \\ {4} & {2} \end{array}\right], \quad B=\left[\begin{array}{rrr} {3} & {2} & {0} \\ {-1} & {-3} & {5} \end{array}\right] $$
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