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IC104 Tutorial 3

This document is a tutorial sheet for the Linear Algebra I course at the Indian Institute of Technology Bhilai, covering various matrix properties and operations. It includes problems on lower and upper triangular matrices, matrix multiplication, finding inverses, solving linear systems, and LU factorization. The exercises aim to enhance understanding of matrix theory and its applications.

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20 views1 page

IC104 Tutorial 3

This document is a tutorial sheet for the Linear Algebra I course at the Indian Institute of Technology Bhilai, covering various matrix properties and operations. It includes problems on lower and upper triangular matrices, matrix multiplication, finding inverses, solving linear systems, and LU factorization. The exercises aim to enhance understanding of matrix theory and its applications.

Uploaded by

annu soni
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Indian Institute of Technology Bhilai

IC104: Linear Algebra I


2021-22-M Instructor: Dr. Raj Kumar Mistri

T UTORIAL S HEET-3

1. Prove that the product of two square lower triangular matrices of same sizes is a lower triangular ma-
trix. Using the property of transpose of a matrix, show that the product of two square upper triangular
matrices of same sizes are upper triangular.

2. Prove that if A, B and C are three matrices (with entries from a field F) of sizes p × q, q × r and r × s,
then
A(BC) = (AB)C.

3. Applying elementary row operations, determine if the inverses of the following matrices exist. If the
inverse exists, then find it.

(a)   (b)
−3 −2 0 2  
 2
 1 0 −1 
. 1 −2 −1
 1 0 1 2 −1 5 6.
2 1 −3 1 5 −4 5

4. Apply the result of Q.3. to find the solution of the following linear system.

−3x1 − 2x2 + 2x4 = 1


2x1 + x2 − x4 = 5
x1 + x3 + 2x4 = −3
2x1 + x2 − 3x3 + x4 = 0

5. Suppose AB = AC, where A is an m × n matrix, and B and C are n × p matrices. If the matrix A is
invertible, show that B = C. Is this true, in general, when A is not invertible.
 
x1
x2 
6. Using Gauss elimination method, solve the matrix equation A⃗x = ⃗b for ⃗x =  x3 , where

x4
   
1 −2 −4 −3 1
 2 −7 −7 −6 7
A=  and ⃗b =   .
−1 2 6 4 0
−4 −1 9 8 3

7. Solve the matrix equation A⃗x = ⃗b in the previous question using LU factorization of the matrix A.

8. Find LU factorization of the matrix


 
1 3 4 0
−3 −6 −7 2 
A= .
 3 3 0 −4
−5 −3 2 9

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