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Chapter 2: Q2.5-11Q (page 93)

Find an LU factorization of the matrices in Exercise 7-16 (with L unit lower triangular). Note that MATLAB will usually produce a permuted LU factorization because it uses partial pivoting for numerical accuracy.

\[\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{ - {\bf{6}}}&{\bf{3}}\\{\bf{6}}&{ - {\bf{7}}}&{\bf{2}}\\{ - {\bf{1}}}&{\bf{7}}&{\bf{0}}\end{array}} \right]\]

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&0&0\\2&1&0\\{ - \frac{1}{3}}&1&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3&{ - 6}&3\\0&5&{ - 4}\\0&0&5\end{array}} \right]\)

Step by step solution

01

Apply the row operation in the given matrix

Let \(A = \left[ {\begin{array}{*{20}{c}}3&{ - 6}&3\\6&{ - 7}&2\\{ - 1}&7&0\end{array}} \right]\).

Apply the row operation to reduce the matrix into the upper triangular matrix.

At row 3, divide row 1 by 3 and add it to row 1, i.e., \({R_3} \to {R_3} + \frac{1}{3}{R_1}\).

\(\left[ {\begin{array}{*{20}{c}}3&{ - 6}&3\\6&{ - 7}&2\\0&5&1\end{array}} \right]\)

At row 2, multiply row 1 with 2 and subtract it from row 2, i.e., \({R_2} \to {R_2} - 2{R_1}\).

\(\left[ {\begin{array}{*{20}{c}}3&{ - 6}&3\\0&5&{ - 4}\\0&5&1\end{array}} \right]\)

02

Apply the row operation in the given matrix

At row 3, subtract row 3 from row 2, i.e., \({R_3} \to {R_3} - {R_2}\).

\(\left[ {\begin{array}{*{20}{c}}3&{ - 6}&3\\0&5&{ - 4}\\0&0&5\end{array}} \right]\)

03

Calculate matrix L using the pivoted column of U

Using matrix \(A\), matrix Lcan be written as shown below:

\(\left[ {\begin{array}{*{20}{c}}3&0&0\\6&5&0\\{ - 1}&5&5\end{array}} \right]\)

Divide column 1 by 3, column 2 by 5, and column 3 by 5.

\(\left[ {\begin{array}{*{20}{c}}1&0&0\\2&1&0\\{ - \frac{1}{3}}&1&1\end{array}} \right]\)

04

Write the product \(LU\)

The product of the lower and upper triangular matrixis

\(LU = \left[ {\begin{array}{*{20}{c}}1&0&0\\2&1&0\\{ - \frac{1}{3}}&1&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3&{ - 6}&3\\0&5&{ - 4}\\0&0&5\end{array}} \right]\).

So, the product \(LU\) is \(\left[ {\begin{array}{*{20}{c}}1&0&0\\2&1&0\\{ - \frac{1}{3}}&1&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3&{ - 6}&3\\0&5&{ - 4}\\0&0&5\end{array}} \right]\).

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Most popular questions from this chapter

If a matrix \(A\) is \({\bf{5}} \times {\bf{3}}\) and the product \(AB\)is \({\bf{5}} \times {\bf{7}}\), what is the size of \(B\)?

Show that if ABis invertible, so is B.

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1鈥�4.

2. \[\left[ {\begin{array}{*{20}{c}}E&{\bf{0}}\\{\bf{0}}&F\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\]

In Exercise 10 mark each statement True or False. Justify each answer.

10. a. A product of invertible \(n \times n\) matrices is invertible, and the inverse of the product of their inverses in the same order.

b. If A is invertible, then the inverse of \({A^{ - {\bf{1}}}}\) is A itself.

c. If \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) and \(ad = bc\), then A is not invertible.

d. If A can be row reduced to the identity matrix, then A must be invertible.

e. If A is invertible, then elementary row operations that reduce A to the identity \({I_n}\) also reduce \({A^{ - {\bf{1}}}}\) to \({I_n}\).

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

38. Use at least three pairs of random \(4 \times 4\) matrices Aand Bto test the equalities \({\left( {A + B} \right)^T} = {A^T} + {B^T}\) and \({\left( {AB} \right)^T} = {A^T}{B^T}\). (See Exercise 37.) Report your conclusions. (Note:Most matrix programs use \(A'\) for \({A^{\bf{T}}}\).
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