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M-1 Question Bank

This document contains a question bank for the course "Linear Algebra & Calculus" with 18 multiple choice questions divided into two units. The questions cover topics like reducing and finding the rank of matrices, computing inverses using elementary row operations, solving systems of equations using various methods, finding eigenvalues and eigenvectors, verifying properties of matrices, and reducing quadratic forms to canonical form through orthogonal transformations. The questions range from lower-order to higher-order thinking skills and align with the specified course outcomes.

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113 views3 pages

M-1 Question Bank

This document contains a question bank for the course "Linear Algebra & Calculus" with 18 multiple choice questions divided into two units. The questions cover topics like reducing and finding the rank of matrices, computing inverses using elementary row operations, solving systems of equations using various methods, finding eigenvalues and eigenvectors, verifying properties of matrices, and reducing quadratic forms to canonical form through orthogonal transformations. The questions range from lower-order to higher-order thinking skills and align with the specified course outcomes.

Uploaded by

scs150831
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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COURSE NAME: Linear Algebra & Calculus

Question Bank
S.No Question RBT CO Marks
level
UNIT-I

[ ]
1 1 2 3 0 L3 1 5
2 4 3 2
Reduce the matrix into Echelon form and find its
3 2 1 3
6 8 7 5
rank

[ ]
2 1 2 1 2 L3 1 5
1 3 2 2
Reduce the matrix by reducing it to canonical
2 4 3 4
3 7 5 6
form hence find its rank

( )
3 1 1 3 L3 1 5
Compute the inverse of the matrix 1 3 −3 by elementary
−2 −4 −4
row operations
4 Test for consistency and hence solve them L3 1 5
x + y + z=6 , x− y +2 z=5 , 3 x + y + z=8 ,2 x−2 y +3 z=7

5 Solve the system of equations L3 1 5


x + y−3 z +2 w=0 , 2 x− y +2 z−3 w=0 , 3 x−2 y + z−4 w=0 ,−4 x+ y−3 z +w=0
6 Show that the only real number λ for which the system L3 1 5
x +2 y+ 3 z= λx ; 3 x+ y+ 2 z= λy ; 2 x+ 3 y+ z =λz has non-zero
solution is 6 and solve them when λ=6
7 Solve the system of equations L3 1 5
3 x+ y+ 2 z=3 ,2 x−3 y−z=−3 , x +2 y + z=4 using Gauss-
Elimination method

8 Using Jacobi’s iteration method solve the system of equations L3 1 5


10 x+ 2 y + z=9 , x +10 y−z=−22 ,−2 x+3 y +10 z=22

9 Using Gauss- Seidel iteration method solve the system of L3 1 5


equations 27 x +6 y−z=85 ,6 x +15 y +2 z=72 , x + y +54 z=110.
UNIT-II
10 Find the eigen values and corresponding eigen vectors of the L3 2 5

( )
8 −6 2
matrix −6 7 −4
2 −4 3
11 Find the Eigen values and Eigen vectors of the matrix L3 2 5

[ ]
6 −2 2
−2 3 −1
2 −1 3

12 Prove that the sum of the eigen values of a square matrix is equal L2 2 5
to its trace and the product of the eigen values is equal to its
determinant
13 Prove that the Eigen values of A−1 are the reciprocals of the L2 2 5
Eigen values of the non singular matrix A.

[ ]
14 1 2 −1 L2 2 5
Verify Cayley- Hamilton theorem for A= 2 1 −2 and hence
2 −2 1
−1
find A ∧A 4

[ ]
15 8 −8 2 L3 2 10
Diagonalize the matrix A= 4 −3 −2
3 −4 1

[ ]
16 1 0 −1 L3 2 10
Diagonalise the matrix A= 1 2 1 and hence calculate A 4
2 2 3
17 Reduce the quadratic form 3 x 2+5 y 2 +3 z 2−2 xy−2 yz +2 zx to L3 2 10
canonical form by orthogonal transformation

18 Reduce the Quadratic form L3 2 10


2 2 2
6 x 1+ 3 x 2 +3 x 3−4 x1 x 2−2 x 2 x 3+ 4 x 1 x 3 to canonical by orthogonal
transformation hence find its nature, rank, index and signature

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