Previous Year Questions: Linear Algebra

(2008-23)

Vector Space

1. Let S be a non-empty set and let V denote the set of all functions from S into R.
Show that V is vector space with respect to the vector addition (𝑓 + 𝑔)(𝑥) =
𝑓(𝑥) + 𝑔(𝑥) and scalar multiplication (𝑐. 𝑓)(𝑥) = 𝑐𝑓(𝑥)
2. Find the dimension of the subspace of 𝑅 spanned by the set
{(1, 0, 0, 0) (0, 1, 0, 0) (1, 2,0, 1) (0, 0, 0, 1)}. Hence find a basis for the subspace. (15)
3. Prove that the set V of the vectors (𝑥 , 𝑥 , 𝑥 , 𝑥 ) in 𝑅 satisfy the equation 𝑥 + 𝑥 +
𝑥 + 𝑥 = 0 and 2𝑥 + 3𝑥 − 𝑥 + 𝑥 = 0 is a subspace of 𝑅 . What is the
dimension of this subspace? Find one of its bases. (12)
4. Prove that the set V of all 3 × 3 real symmetric matrices form a linear subspace of
the space of all 3 × 3 real matrices. What is the dimension of this subspace? Find
at least of the bases for V.
5. In the space 𝑅 . Determine whether or not the set {𝑒 – 𝑒 , 𝑒 – 𝑒 , … . . , 𝑒 – 𝑒 ,𝑒 −
𝑒 } is linearly independent.
6. Let T be a linear transformation from a vector space V over real’s into V such that
𝑇– 𝑇 = 𝐼. Show that T is invertible.
7. Show that the subspaces of 𝑅 spanned by two sets of vectors
{(1, 1, −1), (1, 0, 1)} 𝑎𝑛𝑑 {(1, 2, −3), (5, 2, 1)} are identical. Also find the dimension of
this subspace.
8. Prove or disapprove the following statement: if 𝐵 = {𝑏 , 𝑏 , 𝑏 , 𝑏 , 𝑏 } is a basis for 𝑅
and V is a two-dimensional subspace of 𝑅 , then V has a basis made of two
members of B.

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By Avinash Singh, Ex IES, B. Tech IITR
9. Let V be the vector space of all 2 × 2 matrices over the field of real numbers. Let W
be the set consisting of all matrices with zero determinant. Is W a subspace of V?
Justify your answer. (8)
10. Show that the vectors 𝑋 = (1, 1 + 𝑖, 𝑖), 𝑋 = (𝑖, −𝑖, 1 – 𝑖) and 𝑋 =
(0, 1 – 2𝑖, 2 – 𝑖) in 𝐶 are linearly independent over the field of real numbers but are
linearly dependent over the field of complex numbers.
11. Let V and W be the following subspaces of 𝑅 ∶ 𝑉 = {(𝑎, 𝑏, 𝑐, 𝑑) ∶ 𝑏 – 2𝑐 + 𝑑 = 0}
and 𝑊 = {(𝑎, 𝑏, 𝑐, 𝑑): 𝑎 = 𝑑, 𝑏 = 2𝑐}. Find a basis and the dimension of 𝑉, 𝑊, 𝑉 ∩ 𝑊.
12. The vectors 𝑉 = (1, 1, 2, 4), 𝑉 = (2, −1, −5, 2), 𝑉 = (1, −1, −4, 0) and 𝑉 =
(2, 1, 1, 6) are linearly independent. Is it true? Justify your answer.
13. Find the dimension of the subspace of 𝑅 , spanned by the set
{(1, 0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0, 1)}. Hence find its basis.
14. If 𝑤 = {(𝑥, 𝑦, 𝑧)|𝑥 + 𝑦 – 𝑧 = 0}, 𝑤 = {(𝑥, 𝑦, 𝑧)|3𝑥 + 𝑦 – 2𝑧 = 0}, 𝑤 =
{(𝑥, 𝑦, 𝑧)|𝑥 – 7𝑦 + 3𝑧 = 0} , then find 𝑑𝑖𝑚(𝑤 ∩ 𝑤 ∩ 𝑤 ) and 𝑑𝑖𝑚(𝑤 + 𝑤 ).
15. Suppose U and W are distinct four-dimensional subspaces of a vector space V,
when dim 𝑉 = 6. Find the possible dimensions of subspace 𝑈 ∩ 𝑊.
16. Consider the set V of all 𝑛 × 𝑛 real magic squares. Show that V is a vector space
over R. Give examples of two distinct 2 × 2 magic squares.
17. Show that 𝑆 = {(𝑥, 2𝑦, 3𝑥): 𝑥, 𝑦 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠} is a subspace of 𝑅 (𝑅). Find two
bases of S. Also find the dimension of S.
18. Provet that any set of n linearly independent vectors in a vector space V of
dimension n constitutes a basis for V. (10, 2022)
19. Let 𝑣 = (2, −1,3,2), 𝑣 = (−1,1,1, −3) 𝑎𝑛𝑑 𝑣 = (1,1,9, −5) be three vectors of the
space 𝐼𝑅 . Does (3, −1,0, −1) ∈ span {𝑣 , 𝑣 , 𝑣 } ? Justify your answer. (10, 2023)

Linear Transformation

1. Show that 𝐵 = {(1, 0, 0), (1, 1, 0), (1, 1, 1)} is a basis of 𝑅 . Let 𝑇: 𝑅 → 𝑅 be a linear
transformation such that 𝑇(1, 0, 0) = (1, 0, 0), 𝑇(1, 1, 0) = (1, 1, 1) and 𝑇(1, 1, 1) =
(1, 1, 0). Find 𝑇(𝑥, 𝑦, 𝑧). (15)
2. Let 𝛽 = {(1, 1, 0) (1, 0, 1), (0, 1, 1)} and 𝛽’ = {(2, 1,1), (1, 2, 1), (−1, 1, 1)} be the two
ordered bases of 𝑅 . Then find a matrix representing the linear transformation 𝑇 ∶

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 𝑅 → 𝑅 which transforms 𝛽 into 𝛽’. Use this matrix representation to find 𝑇(𝑥),
where 𝑥 = (2,3, 1).
3. Let 𝐿: 𝑅 → 𝑅 be a linear transformation defined by 𝐿 (𝑥 , 𝑥 , 𝑥 , 𝑥 ) =
(𝑥 + 𝑥 − 𝑥 − 𝑥 , 𝑥 − 𝑥 , 𝑥 – 𝑥 ). Then, find the rank and nullity of L. Also,
determine null space and range space of L. (20)

4. What is the null space of the differential transformation ∶ 𝑃 → 𝑃 where 𝑃 is

the space of all polynomials of degree ≤ 𝑛 over the real numbers? What is the null
space of the second derivatives as a transformation of 𝑃 ? What is the null space
of the 𝑘 derivative 𝑃 ? (12)
4 2 1
5. Let 𝑀 = . Find the unique linear transformation: 𝑅 → 𝑅 , so that M is
0 1 3
the matrix of T with respect to the basis 𝛽 = {𝑣 = (1, 0, 0), 𝑣 = (1, 1, 0), 𝑣 =
(1, 1, 1)} 𝑜𝑓 𝑅 𝑎𝑛𝑑 𝛽’ = {𝑤 = (1, 0), 𝑤 = (1, 1)} of 𝑅 . Also find 𝑇(𝑥, 𝑦, 𝑧). (20)
6. Find the nullity and a basis of the null space of the linear transformation 𝐴 ∶ 𝑅 →
0 1 −3 −1
1 0 1 1
𝑅 , given by the matrix 𝐴 = .
3 1 0 2
1 1 −2 0
7. Show that the vectors (1, 1, 1), (2, 1, 2) 𝑎𝑛𝑑 (1, 2, 3) are linearly independent in 𝑅 . Let
𝑅 → 𝑅 be a linear transformation defined by 𝑇(𝑥, 𝑦, 𝑧) =
(𝑥 + 2𝑦 + 3𝑧, 𝑥 + 2𝑦 + 5𝑧, 2𝑥 + 4𝑦 + 6𝑧). Show that the images of above vectors
under T are linearly dependent. Give the reason for the same. (10)
8. Let 𝑇∶ 𝑅 → 𝑅 be the linear transformation defined by
𝑇(𝛼, 𝛽, 𝛾) = (𝛼 + 2𝛽 − 3𝛾, 2𝛼 + 5𝛽 − 4𝛾, 𝛼 + 4𝛽 + 𝛾). Find a basis and the
dimension of the image of T and the kernel of T. (12)
9. Consider the mapping 𝑓: 𝑅 → 𝑅 𝑏𝑦 𝑓(𝑥, 𝑦) = (3𝑥 + 4𝑦, 2𝑥 – 5𝑦). Find the matrix 𝐴
relative to the basis (1, 0), (0, 1) and the matrix 𝐵 relative to the basis (1, 2), (2, 3).
10. Let 𝑃 denote the vector space of all real polynomials of degree at most n and
𝑇 ∶ 𝑃 → 𝑃 be linear transformation given by 𝑇(𝑓(𝑥)) = ∫ 𝑝(𝑡)𝑑𝑡 , 𝑝(𝑥) 𝜖 𝑃 . Find
the matrix of T with respect to the bases {1, 𝑥, 𝑥 } 𝑎𝑛𝑑 {1, 𝑥, 1 + 𝑥 , 1 + 𝑥 } of 𝑃 and
𝑃 respectively. Also find the null space of T. (10)
11. Let V be an n-dimensional vector space and 𝑇 ∶ 𝑉 → 𝑉 be an invertible linear
operator. If 𝛽 = {𝑋 , 𝑋 , … 𝑋 }is a basis of V, show that 𝛽’ = {𝑇𝑋 , 𝑇𝑋 … 𝑇𝑋 } is also
a basis of V. (8)

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12. Let 𝑉 = 𝑅 and 𝑇 𝜖 𝐴(𝑉), for all 𝑎 𝜖 𝐴(𝑉), be defined by
𝑇(𝑎 , 𝑎 , 𝑎 ) = (2𝑎 + 5𝑎 + 𝑎 , −3𝑎 + 𝑎 − 𝑎 , 𝑎 + 2𝑎 + 3𝑎 ). What is the
matrix T relative to the basis 𝑉 = (1, 0, 1), 𝑉 = (−1, 2, 1), 𝑉 = (3, −1, 1) ?
13. If 𝑀2(𝑅) is space of real matrices of order 2 × 2 and 𝑃2(𝑥) is the space of real
polynomials of degree at most 2, then find the matrix representation of 𝑇: 𝑀2(𝑅) → 𝑃2(𝑥)
𝑎 𝑏
such that 𝑇 = = 𝑎 + 𝑏 + 𝑐 + (𝑎 – 𝑑)𝑥 + (𝑏 + 𝑐)𝑥 , with respect to the
𝑐 𝑑
standard bases of 𝑀 (𝑅) and 𝑃 (𝑥). Further find the null space of T.
14. If 𝑇: 𝑃 (𝑥) → 𝑃 (𝑥) is such that 𝑇 𝑓(𝑥) = 𝑓(𝑥) + 5 ∫ 𝑓(𝑡)𝑑𝑡 , then choosing
{1, 1 + 𝑥, 1 – 𝑥 } and {1, 𝑥, 𝑥 , 𝑥 } as bases of 𝑃 (𝑥) and 𝑃 (𝑥) respectively, find the
matrix of T. (10)
1 −1 2
15. If 𝐴 = −2 1 −1 is the matrix representation of a linear transformation
1 2 3
𝑇: 𝑃 (𝑥) → 𝑃 (𝑥) with respect to the bases {1 – 𝑥, 𝑥(1 – 𝑥), 𝑥(1 + 𝑥)} 𝑎𝑛𝑑 {1, 1 +
𝑥, 1 + 𝑥 } , then find T. (18)
1 2 3 1
16. Consider the matrix mapping 𝐴: 𝑅 → 𝑅 , where 𝐴 = 1 3 5 −2 . Find a
3 8 13 −3
basis and dimension of the image of A and those of kernel A.
17. Let 𝑇 ∶ 𝑅 → 𝑅 be a linear map such that 𝑇(2, 1) = (5, 7) and 𝑇(1, 2) = (3, 3).
If A is the matrix corresponding to T with respect to the standard bases (𝑒 , 𝑒 ),
then find Rank (A).
18. Let 𝑀 (𝑅)be the vector space of all 2×2 real matrices. Let 𝐵=
1 −1
. Suppose 𝑇 ∶ 𝑀 (𝑅) → 𝑀 (𝑅) is a linear transformation defined by 𝑇 (𝐴) =
−4 4
𝐵𝐴. Find the rank and nullity of T. Find a matrix A which maps to the null matrix.
19. Let F be a subfield of complex numbers and T is a function from 𝐹 → 𝐹
defined by 𝑇(𝑥 , 𝑥 , 𝑥 ) = (𝑥 + 𝑥 + 3𝑥 , 2𝑥 − 𝑥 , −3𝑥 + 𝑥 − 𝑥 ).What are the
conditions on a, b, c, such that ( 𝑎, 𝑏, 𝑐 ) be in the null space of T ? Find the nullity
of T.
20. Find the matrix associated with the linear operator on 𝑉 (𝑅) defined by
𝑇(𝑎, 𝑏, 𝑐) = (𝑎 + 𝑏, 𝑎 − 𝑏, 2𝑐), with respect to ordered basis 𝐵 = {(0,1,1), (1,0,1), (1,1,0)}.
(10,2021)

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By Avinash Singh, Ex IES, B. Tech IITR
 1
1 1
21. Let 𝑇: 𝑅 → 𝑅 be a linear transformation such that 𝑇 = 2 and 𝑇 =
0 1
3
−3
2
2 . Find 𝑇 . (10,2022).
4
8
22. Find the rank and nullity of the linear transformation 𝑇: 𝐼𝑅 → 𝐼𝑅 given by
𝑇(𝑥, 𝑦, 𝑧) = (𝑥 + 𝑧, 𝑥 + 𝑦 + 2𝑧, 2𝑥 + 𝑦 + 3𝑧). (10, 2023)
23. If the matrix of linear transformation 𝑇: 𝐼𝑅 → 𝐼𝑅 relative to the basis
1 1 2
{(1,0,0), (0,1,0), (0,0,1)} is −1 2 1 , then find the matrix of T relative to the basis
0 1 2
{(1,1,1), (0,1,1), (0,0,1)}. (15, 2023)

Matrices

1. Show that the matrix 𝐴 is invertible if and only if the 𝑎𝑑𝑗 (𝐴 ) is invertible. Hence
find |𝑎𝑑𝑗 (𝐴)|. (12)
2. Let A be a non-singular matrix. Show that if 𝐼 + 𝐴 + 𝐴 + … + 𝐴 = 0, then
𝐴 = 𝐴 .(15)
3. Find a Hermitian and skew-hermitian matrix each, whose sum is the matrix
2𝑖 3 −1
1 2 + 3𝑖 2
−𝑖 + 1 4 5𝑖
4. Find a 2 × 2 real matrix A which is both orthogonal and skew-symmetric. Can
there exist a 3 × 3 real matrix which is both orthogonal and skew-symmetric?
Justify your answer. (20)
26 −2 2
5. If 𝜆 , 𝜆 , 𝜆 are the Eigen values of matrix 𝐴 = 2 21 4 , show that
44 2 28
𝜆 + 𝜆 + 𝜆 ≤ √1949. (12)
6. Let A and B be 𝑛 × 𝑛 matrices over reals. Show that 𝐼 − 𝐵𝐴 is invertible if 𝐼 – 𝐴𝐵 is
invertible. Deduce that 𝐴𝐵 𝑎𝑛𝑑 𝐵𝐴 have same Eigen values. (20)
7. Let A be a non-singular 𝑛 × 𝑛, square matrix. Show that 𝐴. (𝑎𝑑𝑗𝐴) = |𝐴|𝐼 . Hence
show that |𝑎𝑑𝑗(𝑎𝑑𝑗𝐴)| = |𝐴|( – )
.
8. Let 𝜆 , 𝜆 , … , 𝜆 be the Eigen values of a 𝑛 × 𝑛 square matrix A with corresponding
Eigen vectors 𝑋 , 𝑋 , … , 𝑋 . If B is a matrix similar to A, show that the Eigen values
of B are same as that of A. Also find the relation between the Eigen vectors of B
and Eigen vectors of A. (10)

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 2 −2 2
9. Let 𝐴 = 1 1 1 and C be a non-singular matrix of order 3 × 3. Find the Eigen values
1 3 −1
of the matrix 𝐵 where 𝐵 = 𝐶 𝐴𝐶. (10)
10. If 𝜆 is a characteristic root of a non-singular matrix A then prove that |𝐴| /𝜆 is
a characteristic root of 𝐴𝑑𝑗. 𝐴
1 𝑖 2+𝑖
11. Let 𝐻 = −𝑖 2 1 − 𝑖 be a Hermitian matrix. Find a non-singular matrix
2−𝑖 1+𝑖 2
𝑃 such that 𝐷 = 𝑃 𝐻𝑃 is diagonal.
12. Let A be a square matrix and 𝐴∗ be its adjoint, show that the Eigen values of
matrices 𝐴𝐴∗ and 𝐴∗ 𝐴 are real. Further show that 𝑇𝑟𝑎𝑐𝑒 (𝐴𝐴∗ ) = 𝑇𝑟𝑎𝑐𝑒 (𝐴∗ 𝐴).
1 1 1
13. Let 𝐴 = 1 𝜔 𝜔 where 𝜔 (≠ 1) is a cube root of unity. If 𝜆1, 𝜆2, 𝜆3, denote the
1 𝜔 𝜔
eigenvalues of 𝐴 , show that |𝜆 | + |𝜆 | + |𝜆 | ≤ 9.
1 2 3 4 5
⎡2 3 5 8 12⎤
⎢ ⎥
14. Find the rank of the matrix 𝐴 = ⎢3 5 8 12 17⎥.
⎢3 5 8 17 23⎥
⎣8 12 17 23 30⎦
15. Let A be a Hermitian matrix having all distinct eigenvalues 𝜆 , 𝜆 , … , 𝜆 . If
𝑋 , 𝑋 , … 𝑋 are corresponding Eigen vectors then show that the 𝑛 × 𝑛 matrix C
whose 𝑘 column consists of the vector 𝑋 is non singular. (8)
16. Using elementary row or column operations, find the rank of the matrix
0 1 −3 −1
0 0 1 1
3 1 0 2
1 1 −2 0
1 4
17. Verify Cayley – Hamilton theorem for the matrix 𝐴 = and hence find its
2 3
inverse. Also find the matrix representation 𝐴 – 4𝐴 – 7𝐴 + 11𝐴 – 𝐴 – 10𝐼.
−2 2 −3
18. Let 𝐴 = 2 1 −6 . Find the Eigen values of A and the corresponding
−1 −2 0
Eigen vectors.
19. Prove that Eigen values of a unitary matrix have absolute value 1.
20. Reduce the following matrix to row echelon form and hence find its
1 2 3 4
2 1 4 5
rank:
1 5 5 7
8 1 14 17

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 1 0 0
21. If matrix 𝐴 = 1 0 1 , then find 𝐴 .
0 1 0
1 1 3
22. Find the Eigen values and Eigen vectors of the matrix 1 5 1 .
3 1 1
1 2 1
23. Using elementary row operations, find the inverse of 𝐴 = 1 3 2 .
1 0 1
1 1 3
24. If 𝐴 = 5 2 6 , then find 𝐴 + 3𝐴 – 2𝐼.
−2 −1 −3
25. Using elementary row operations, find the condition that the linear equations
have a solution: 𝑥 – 2𝑦 + 𝑧 = 𝑎; 2𝑥 + 7𝑦 – 3𝑧 = 𝑏; 3 𝑥 + 5𝑦 – 2𝑧 = 𝑐 . (7)
1 1 0
26. If 𝐴 = 1 1 0 , then find the Eigenvalues and Eigenvectors of 𝐴.
0 0 1
27. Prove that Eigen values of a Hermitian matrix are all real.
2 2
28. Let 𝐴 = . Find a non-singular matrix 𝑃 such that 𝑃 𝐴𝑃 is diagonal
1 3
matrix.
29. Show that similar matrices have the same characteristic polynomial.
30. Prove that the distinct non-zero eigen vectors of a matrix are linearly
independent.
31. Let A be a 3 × 2 matrix and B a 2 × 3 matrix. Show that 𝐶 = 𝐴. 𝐵 is a singular
matrix.
32. Let A and B be two orthogonal matrices of same order and 𝑑𝑒𝑡 𝐴 + 𝑑𝑒𝑡 𝐵 = 0,
show that 𝐴 + 𝐵 is a singular matrix.
5 7 2 1
1 1 −8 1
33. Let 𝐴 =
2 3 5 0
3 4 −3 1
a. Find the rank of the matrix A.
b. Find the dimension of the subspace 𝑉 =
𝑥
𝑥
(𝑥 , 𝑥 , 𝑥 , 𝑥 ) ∈ 𝑅 𝐴 𝑥 = 0
𝑥
1 0 0
34. 𝐴 = 1 0 1 , State the Cayley-Hamilton theorem. Use this theorem to find
0 1 1
𝐴 .

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35. Define an 𝑛 × 𝑛 matrix as 𝐴 = 𝐼 − 2𝑢 𝑢 , where u is a unit column vector
(i) Examine if A is symmetric.
(ii) Examine if A is orthogonal.
(iii) Show that 𝑡𝑟𝑎𝑐𝑒 (𝐴) = 𝑛 − 2
1/3
(iv) Find 𝐴 × when 𝑢 = 2/3
2/3
1 −1 1
36. If 𝐴 = 2 −1 0 , then show that 𝐴 = 𝐴 (𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑓𝑖𝑛𝑑𝑖𝑛𝑔 𝐴 ). (10, 2021)
1 0 0
37. Prove that the eigen vectors, corresponding to two distinct eigne values of a
real symmetric matrix, are orthogonal. (8, 2021)
38. For two square matrices A and B of order 2, show that 𝑡𝑟𝑎𝑐𝑒 (𝐴𝐵) = 𝑡𝑟𝑎𝑐𝑒 (𝐵𝐴).
Hence show that 𝐴𝐵 − 𝐵𝐴 ≠ 𝐼 , 𝑤ℎ𝑒𝑟𝑒 𝐼 is identity matrix is order 2. (7, 2021)
39. Reduce the following matrix to a row reduced echelon form and hence, also,
find its rank.
1 3 2 4 1
0 0 2 2 0
(10, 2021)
2 6 2 6 2
3 9 1 10 6
40. Find the eigen values and corresponding eigen vectors of the matrix 𝐴 =
0 −𝑖
, over the complex-number field. (10, 2021)
𝑖 0
𝑥
41. Let the set 𝑃 = 𝑦 𝑥 − 𝑦 − 𝑧 = 0 𝑎𝑛𝑑 2𝑥 − 𝑦 + 𝑧 = 0} be the collection of
𝑧
vectors of a vector space 𝑅 (𝑅).Then
a. Prove that P is subspace of 𝑅
b. Find the basis and dimension of P. (10+10)
42. Find a linear map 𝑇: 𝑅 → 𝑅 which rotates each vector of 𝑅 by an angle 𝜃.
Also prove that for 𝜃 = , T has no eigenvalue in R. (15, 2022)
1 0 0
43. Let 𝐴 = 1 0 1
0 1 1
i. Verify the Cayley-Hamilton theorem for the matrix A.
ii. Show that 𝐴 = 𝐴 + 𝐴 − 𝐼 𝑓𝑜𝑟 𝑛 ≥ 3, where I is the identity
matrix of order 3. Hence find 𝐴 . (10+10, 2023)

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By Avinash Singh, Ex IES, B. Tech IITR
 1 2 −1 0
−1 3 0 −4
44. Find the rank of the matrix 𝐴 = by reducing it to row reduced
2 1 3 −2
1 1 1 −1
echelon form. (15, 2023)

Solution of System of Linear Equations

1 0 −1 𝑥 2
1. Let = 3 4 5 , 𝑋 = 𝑦 , 𝐵 = 6 . Solve the system of equations given by
0 6 7 𝑧 5
𝐴𝑋 = 𝐵. Using the above, also solve the system of equations 𝐴 𝑋 = 𝐵 where 𝐴
denotes the transpose matrix of A.(10)
2. Find the dimension and a basis for the space W of all solutions of the following
𝑥 + 2𝑥 + 3𝑥 – 2𝑥 + 4𝑥 = 0
homogeneous system using matrix notation: 2𝑥 + 4𝑥 + 8𝑥 + 𝑥 + 9𝑥 = 0
3𝑥 + 6𝑥 + 13𝑥 + 4𝑥 + 14𝑥 = 0

1 3 1
3. Find the inverse of matrix 𝐴 = 2 −1 7 by using elementary row operations.
3 2 −1
Hence solve the system of linear equations
𝑥 + 3𝑦 + 𝑧 = 10
2𝑥 – 𝑦 + 7𝑧 = 12
3𝑥 + 2𝑦 – 𝑧 = 4
4. Investigate the values of 𝜆 and µ so that the equations 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + 3𝑧 =
10, 𝑥 + 2𝑦 + 𝜆𝑧 = µ have
a. No solution
b. Unique solution
c. An infinite number of solutions
5. Consider the following system of equation in x, y, z
𝑥 + 2𝑦 + 2𝑧 = 1
𝑥 + 𝑎𝑦 + 3𝑧 = 3
𝑥 + 11𝑦 + 𝑎𝑧 = 𝑏
a. For which values of ‘a’ does that system have a unique?
b. For which of values (a, b) does the system have more than one solution?

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By Avinash Singh, Ex IES, B. Tech IITR
 1 2 1 2 1 1
6. If 𝐴 = 1 −4 1 and 𝐵 = 1 −1 0 . Then, show that 𝐴𝐵 = 6𝐼 . Use this
3 0 −3 2 1 −1
result to solve the following system of equations: 2𝑥 + 𝑦 + 𝑧 = 5; 𝑥 – 𝑦 =
0; 2𝑥 + 𝑦 – 𝑧 = 1 ;
7. For the system of linear equations
𝑥 + 3𝑦 – 2𝑧 = −1; 5𝑦 + 3𝑧 = −8 ; 𝑥 − 2𝑦 – 5𝑧 = 7,
a. determine the following statements, which are true or false:
i. The system has no solution
ii. The system has unique solution
iii. The system has infinitely many solutions
1 0 2 1 0 2
8. Let 𝐴 = 2 −1 3 and 𝐵 = 2 −1 3
4 1 8 4 1 8
a. Find 𝐴𝐵
b. Find 𝑑𝑒𝑡(𝐴) and 𝑑𝑒𝑡(𝐵)
c. Solve the system of linear equations 𝑥 + 2𝑧 = 3; 2𝑥 − 𝑦 + 3𝑧 = 3; 4𝑥 + 𝑦 +
8𝑧 = 14.
9. Find all solution to the following system of equations by row reduced method:

𝑥 + 2𝑥 − 𝑥 = 2
2𝑥 + 3𝑥 + 5𝑥 = 5 (15, 2022)
−𝑥 − 3𝑥 + 8𝑥 = −1

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By Avinash Singh, Ex IES, B. Tech IITR