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Exercises $42-44$ show how to use the condition number of a matrix $A$ to estimate the accuracy of a computed solution of $A \mathbf{x}=\mathbf{b}$ . If the entries of $A$ and $\mathbf{b}$ are accurate to about $r$ significant digits and if the condition number of $A$ is approximately $10^{k}$ (with $k$ a positive integer), then the computed solution of $A \mathbf{x}=\mathbf{b}$ should usually be accurate to at least $r-k$ significant digits.[IM ] Repeat Exercise 42 for the matrix in Exercise 10 .

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Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

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University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

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07:20

Exercises $42-44$ show how…

06:49

01:43

In Exercises $1-4,$ the ma…

07:34

In Exercises $37-40$ , use…

00:25

[M] Compute an SVD of each…

00:38

03:28

[M] Use as many columns of…

06:35

Exercises $22-26$ provide …

05:13

in Problem 43. We have these metrics square metrics five by five We call it metrics A. We will take any random victor X where X equals B. Then X must be in are five. Then we will take X equals. For example one zero minus five eight minus two. Then the multiplication off A and X gives the victor B and after multiplication multiplying road by column Road by column It's row by the vector. It's true by the victor, for example, we have five plus zero minus five equals zero Lost seven, multiplied by eight equals 56 minus 18 equals 38 and so on to get 76. 54 buying a 73 78. Then we will use a metrics program. Have you used Excel to get X for example? Now, if we don't know X, we don't know this Victor. We will use the program to get to solve for X here and call it X one from Excel Way Get X one equals using the inverse off equals in verse about to blow it by B and it gives this ends These victim for 11. This milk places it gives one 000 until four in the 11. This small place and it gives minus a point or or or or until without six in the 11th, this small place. Then we have minus 4.999 until mine in the elefant, This small place. And we got eight point. Oh, and finally of minus two point or Oh, it approximately equals one minus zero, which is zero minus 4.5. It, um, minus two. This is accurate. It's accurate. Toe nine, this woman places because if we take the Kirsty to 10 decimal places, this will give us here zeros until we got one. In the end, which is not accurate to zero, that it's accurate toe mind this small places. And this is the final answer off our group.

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In mathematics, the absolute value or modulus |x| of a real number x is its …

Exercises $42-44$ show how to use the condition number of a matrix $A$ to es…

In Exercises $1-4,$ the matrix $A$ is followed by a sequence $\left\{\mathbf…

In Exercises $37-40$ , use a matrix program to find the eigenvalues of the m…

[M] Compute an SVD of each matrix. Report the final matrix entries accurate …

[M] Use as many columns of $A$ as possible to construct a matrix $B$ with th…

Exercises $22-26$ provide a glimpse of some widely used matrix factorization…

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Use Cramer’s rule to compute the solutions of the systems in Exercises 1–6.<…

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Unless otherwise specified, assume that all matrices in these exercises are …

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Find formulas for X, Y, and Z in terms of A, B, and C, and justify your calc…

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In Exercises 1–9, assume that the matrices are partitioned conformably for b…

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Find an LU factorization of the matrices in Exercises $7-16$ (with $L$ unit …

14:39

Let $\mathbf{v}_{1}=\left[\begin{array}{r}{2} \\ {-8} \\ {6}\end{array}\righ…

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Prove Theorem 2$(\mathrm{b})$ and 2$(\mathrm{c}) .$ Use the row-column rule.…

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Suppose an $m \times n$ matrix $A$ has $n$ pivot columns. Explain why for ea…

If $C$ is $6 \times 6$ and the equation $C \mathbf{x}=\mathbf{v}$ is consist…

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Prove the second part of Theorem $6 :$ Let $\mathbf{w}$ be any solution of $…

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