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Chapter 3: Problem 15

$$ \left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right] $$

Short Answer

Expert verified
The inverse of the given lower triangular matrix is: \[ \left[\begin{array}{ccc} 1 & -1 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array}\right] \]

Step by step solution

01

Write the given matrix alongside the identity matrix

Write down both the given matrix and the identity matrix of the same size (3x3) as an augmented matrix: \[ \left[\begin{array}{@{}ccc|ccc@{}} 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{array}\right] \]
02

Perform row operations

Since the given matrix is a lower triangular matrix with nonzero diagonal entries, the row operations needed to get the identity matrix on the left side will be subtraction. Perform the following row operations: - Row 1 = Row 1 - Row 2 - Row 3 \[ \left[\begin{array}{@{}ccc|ccc@{}} 1 & 0 & 0 & 1 & -1 & -1 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{array}\right] \] - Row 2 = Row 2 - Row 3 \[ \left[\begin{array}{@{}ccc|ccc@{}} 1 & 0 & 0 & 1 & -1 & -1 \\ 0 & 1 & 0 & 0 & 1 & -1 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{array}\right] \]
03

Write down the inverse matrix

After performing the row operations, we have obtained the identity matrix on the left side. The matrix on the right side is the inverse of the given matrix: \[ \left[\begin{array}{@{}ccc@{}} 1 & -1 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array}\right] \]

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Key Concepts

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

Lower Triangular Matrix
A lower triangular matrix is a special type of square matrix where all the elements above the principal diagonal are zero. This means that only the diagonal and the elements below it can have non-zero values. For example, in a 3x3 lower triangular matrix, the top row could be
  • x, 0, 0
  • where x is a non-zero value.
  • The second row might be y, z, 0,
  • with y and z as non-zero values.
  • And the last row could be a, b, c,
  • with a, b, and c as non-zero values.

Lower triangular matrices are especially useful in solving systems of linear equations, simplifying operations, and understanding matrix properties. They are often easy to invert when the diagonal elements are non-zero, as the upper part is already in a simplified form, making calculations straightforward.
Elementary Row Operations
Elementary row operations are used to manipulate the rows of a matrix. These operations help in transforming a matrix to another form, such as row echelon form or reduced row echelon form. There are three basic types of row operations:
  • Row Swap: Interchanging two rows.
  • Row Multiplication: Multiplying all elements of a row by a non-zero scalar.
  • Row Addition/Subtraction: Adding or subtracting the elements of one row to or from another row.

These operations play a crucial role in the process of matrix inversion as they aid in simplifying the matrix to the identity matrix, allowing the reveal of the inverse matrix on the other side of augmented matrices. A good understanding of these operations is essential for solving a variety of linear algebra problems.
Identity Matrix
The identity matrix is a special matrix that acts as the multiplicative identity in matrix algebra, similar to how the number 1 behaves in arithmetic. It is a square matrix with ones on the diagonal and zeros elsewhere. For instance, a 3x3 identity matrix looks like this:
  • 1, 0, 0
  • 0, 1, 0
  • 0, 0, 1

When a matrix is multiplied by the identity matrix, it remains unchanged. This unique property makes the identity matrix extremely important in solving linear equations and finding inverse matrices. In matrix inversion, the goal is often to perform row operations until the left half of an augmented matrix becomes the identity matrix. At this stage, the right half of the augmented matrix becomes the inverse of the original matrix.

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Most popular questions from this chapter

Is it possible for \(a 2 \times 3\) matrix to equal a \(3 \times 2\) matrix? Explain.

If \(A\) and \(B\) are \(2 \times 3\) matrices and \(A=B\), what can you say about \(A-B ?\) Explain.

Factory Location \(^{23}\) A manufacturer of electrical machinery is located in a cramped, though low-rent, factory close to the center of a large city. The firm needs to expand, and it could do so in one of three ways: (1) remain where it is and install new equipment, \((2)\) move to a suburban site in the same city, or (3) relocate in a different part of the country where labor is cheaper. Its decision will be influenced by the fact that one of the following will happen: (I) the government may introduce a program of equipment grants, (II) a new suburban highway may be built, or (III) the government may institute a policy of financial help to companies who move into regions of high unemployment. The value to the company of each combination is given in the following payoff matrix. Government's Options \begin{tabular}{|c|c|c|c|} \hline & I & II & III \\ \hline \(\mathbf{1}\) & 200 & 150 & 140 \\ \hline \(\mathbf{2}\) & 130 & 220 & 130 \\ \hline \(\mathbf{3}\) & 110 & 110 & 220 \\ \hline \end{tabular} Manufacturer's Options If the manufacturer judges that there is a \(20 \%\) probability that the government will go with option I, a \(50 \%\) probability that they will go with option II, and a \(30 \%\) probability that they will go with option III, what is the manufacturer's best option?

I Population Movement In 2006, the population of the United States, broken down by regions, was \(55.1\) million in the Northeast, \(66.2\) million in the Midwest, \(110.0\) million in the South, and \(70.0\) million in the West. \({ }^{19}\) The matrix \(P\) below shows the population movement during the period 2006 \(2007 .\) (Thus, \(98.92 \%\) of the population in the Northeast stayed there, while \(0.17 \%\) of the population in the Northeast moved to the Midwest, and so on.) \(\begin{array}{ccll}\text { To } & \text { To } & \text { To } & \text { To } \\\ \text { NE } & \text { MW } & \text { S } & \text { W }\end{array}\) \(P=\begin{aligned}&\text { From NE } \\\&\text { From MW } \\\&\text { From S } \\\&\text { From W }\end{aligned}\left[\begin{array}{llll}0.9892 & 0.0017 & 0.0073 & 0.0018 \\ 0.0010 & 0.9920 & 0.0048 & 0.0022 \\ 0.0018 & 0.0024 & 0.9934 & 0.0024 \\ 0.0008 & 0.0033 & 0.0045 & 0.9914\end{array}\right]\) Set up the 2006 population figures as a row vector. Assuming that these percentages also describe the population movements from 2005 to 2006 , show how matrix inversion and multiplication allow you to compute the population in each region in 2005 . (Round all answers to the nearest \(0.1\) million.)

Decide whether the game is strictly determined. If it is, give the players'optimal pure strategies and the value of the game. $$ \begin{array}{r} \mathbf{B} \\ p & q & r \\ a & {\left[\begin{array}{rrr} 2 & 0 & -2 \\ -1 & 3 & 0 \end{array}\right]} \end{array} $$
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