JEE Main Previous Year Questions (2021-2026): Matrices and Determinants

 Page 1


 
JEE Main Previous Year Questions (2021-2026): 
Matrices and Determinants 
 
(January 2026) 
 
Q1: Let A = and B be two matrices such that A¹ ° ° = 100B + I. Then the sum of 
all the elements of B¹ ° ° is 
Ans: 0 
Sol: 
Decompose Matrix A  
We can write matrix A as the sum of the Identity matrix I and a residual matrix C: 
 
Since C² = 0, C is a nilpotent matrix. Consequently, Cn = 0 for all n = 2. 
Find A¹ ° ° using Binomial Expansion : 
A¹ ° ° = (I + C)¹ ° ° = ¹ ° °C 0I¹ ° ° + ¹ ° °C 1I ? ?C + ¹ ° °C 2I ? 8C² + … + ¹ ° °C 1 0 0C¹ ° ° 
Since C² = C³ = … = C¹ ° ° = 0, the expansion simplifies to : 
A¹ ° ° = ¹ ° °C 0I + ¹ ° °C 1C 
A¹ ° ° = 1 · I + 100 · C 
A¹ ° ° = I + 100C 
Find Matrix B : 
The problem states that A¹ ° ° = 100B + I. Comparing this to our result: 
100B + I = 100C + I 
100B = 100C 
B = C 
Calculate the Sum of Elements of B¹ ° ° : 
Since B = C and we know C² = 0, then:   B² = 0 
Page 2


 
JEE Main Previous Year Questions (2021-2026): 
Matrices and Determinants 
 
(January 2026) 
 
Q1: Let A = and B be two matrices such that A¹ ° ° = 100B + I. Then the sum of 
all the elements of B¹ ° ° is 
Ans: 0 
Sol: 
Decompose Matrix A  
We can write matrix A as the sum of the Identity matrix I and a residual matrix C: 
 
Since C² = 0, C is a nilpotent matrix. Consequently, Cn = 0 for all n = 2. 
Find A¹ ° ° using Binomial Expansion : 
A¹ ° ° = (I + C)¹ ° ° = ¹ ° °C 0I¹ ° ° + ¹ ° °C 1I ? ?C + ¹ ° °C 2I ? 8C² + … + ¹ ° °C 1 0 0C¹ ° ° 
Since C² = C³ = … = C¹ ° ° = 0, the expansion simplifies to : 
A¹ ° ° = ¹ ° °C 0I + ¹ ° °C 1C 
A¹ ° ° = 1 · I + 100 · C 
A¹ ° ° = I + 100C 
Find Matrix B : 
The problem states that A¹ ° ° = 100B + I. Comparing this to our result: 
100B + I = 100C + I 
100B = 100C 
B = C 
Calculate the Sum of Elements of B¹ ° ° : 
Since B = C and we know C² = 0, then:   B² = 0 
Consequently, B¹ ° ° = 0 (the zero matrix). 
 
The sum of all the elements of B
100
 is 0. 
 
Q2: The number of 3 × 2 matrices A, which can be formed using the elements of the set 
{-2, -1, 0, 1, 2} such that the sum of all the diagonal elements of A?A is 5, is 
Ans: 312 
Sol: 
 
sum of diagonal elements of A?A = 5 
a² + c² + e² + b² + d² + f² = 5 
possible values of squares is {0, 1, 4} 
Case 1 : one element is 4,1 element is 1 and four elements are 0 . 
one element is 4 = 2 choices {-2, 2} 
one element is 1 = 2 choices {-1, 1} 
4 element is zero = 1 choice {0} 
Selecting one element 4 out of 6 choice {a², b², c², d², e², f²}, and have 2 choice {-2, 2}, total = 
6C 1 × 2 = 12 
Selecting one element which is 1 from remaining 5 element = 5C 1, and have two choice {-1, 1}, 
total = 5C 1 × 2 = 10 
Selecting 4 element which is zero from remaining 4 elements = 4C 4 = 1 total matrices for case 1 
= 12 × 10 × 1 = 120 
Case 2 : five elements are 1 and one element is 0. 
squares a², b², c², d², e², f² ? {1, 1, 1, 1, 1, 0} 
Page 3


 
JEE Main Previous Year Questions (2021-2026): 
Matrices and Determinants 
 
(January 2026) 
 
Q1: Let A = and B be two matrices such that A¹ ° ° = 100B + I. Then the sum of 
all the elements of B¹ ° ° is 
Ans: 0 
Sol: 
Decompose Matrix A  
We can write matrix A as the sum of the Identity matrix I and a residual matrix C: 
 
Since C² = 0, C is a nilpotent matrix. Consequently, Cn = 0 for all n = 2. 
Find A¹ ° ° using Binomial Expansion : 
A¹ ° ° = (I + C)¹ ° ° = ¹ ° °C 0I¹ ° ° + ¹ ° °C 1I ? ?C + ¹ ° °C 2I ? 8C² + … + ¹ ° °C 1 0 0C¹ ° ° 
Since C² = C³ = … = C¹ ° ° = 0, the expansion simplifies to : 
A¹ ° ° = ¹ ° °C 0I + ¹ ° °C 1C 
A¹ ° ° = 1 · I + 100 · C 
A¹ ° ° = I + 100C 
Find Matrix B : 
The problem states that A¹ ° ° = 100B + I. Comparing this to our result: 
100B + I = 100C + I 
100B = 100C 
B = C 
Calculate the Sum of Elements of B¹ ° ° : 
Since B = C and we know C² = 0, then:   B² = 0 
Consequently, B¹ ° ° = 0 (the zero matrix). 
 
The sum of all the elements of B
100
 is 0. 
 
Q2: The number of 3 × 2 matrices A, which can be formed using the elements of the set 
{-2, -1, 0, 1, 2} such that the sum of all the diagonal elements of A?A is 5, is 
Ans: 312 
Sol: 
 
sum of diagonal elements of A?A = 5 
a² + c² + e² + b² + d² + f² = 5 
possible values of squares is {0, 1, 4} 
Case 1 : one element is 4,1 element is 1 and four elements are 0 . 
one element is 4 = 2 choices {-2, 2} 
one element is 1 = 2 choices {-1, 1} 
4 element is zero = 1 choice {0} 
Selecting one element 4 out of 6 choice {a², b², c², d², e², f²}, and have 2 choice {-2, 2}, total = 
6C 1 × 2 = 12 
Selecting one element which is 1 from remaining 5 element = 5C 1, and have two choice {-1, 1}, 
total = 5C 1 × 2 = 10 
Selecting 4 element which is zero from remaining 4 elements = 4C 4 = 1 total matrices for case 1 
= 12 × 10 × 1 = 120 
Case 2 : five elements are 1 and one element is 0. 
squares a², b², c², d², e², f² ? {1, 1, 1, 1, 1, 0} 
selection of 5 one from a², b², c², d², e², f² = 6C 5 
and for 1 there are two choice for each elements a, b, c, d, e, f = {-1, 1}, total = 2 5 
for 0 only 1 choices 
total matrices of case 2 = 6C 5 × 2 5 = 192 
total number of such matrices = Case 1 + Case 2 
= 120 + 192 = 312 
 
Q3: Let  and B be a matrix such that B (I - A) = I + A. Then 
the sum of the diagonal elements of B
T
B is equal to 
Ans: 3 
Sol: Observing properties of Matrix A 
 
Rearranging the given equation 
B(I - A) = I + A 
B = (I + A)(I - A) ?¹ 
Finding B? 
B? = [(I + A)(I - A) ?¹]? 
B? = ((I - A) ?¹)? (I + A)? 
B? = (I - A?) ?¹ (I + A?) 
Since A? = -A : 
B? = (I + A) ?¹(I - A) 
B?B = [(I + A) ?¹(I - A)] · [(I + A)(I - A) ?¹] 
B?B = (I + A) ?¹(I + A)(I - A)(I - A) ?¹ 
B?B = I · I = I 
Page 4


 
JEE Main Previous Year Questions (2021-2026): 
Matrices and Determinants 
 
(January 2026) 
 
Q1: Let A = and B be two matrices such that A¹ ° ° = 100B + I. Then the sum of 
all the elements of B¹ ° ° is 
Ans: 0 
Sol: 
Decompose Matrix A  
We can write matrix A as the sum of the Identity matrix I and a residual matrix C: 
 
Since C² = 0, C is a nilpotent matrix. Consequently, Cn = 0 for all n = 2. 
Find A¹ ° ° using Binomial Expansion : 
A¹ ° ° = (I + C)¹ ° ° = ¹ ° °C 0I¹ ° ° + ¹ ° °C 1I ? ?C + ¹ ° °C 2I ? 8C² + … + ¹ ° °C 1 0 0C¹ ° ° 
Since C² = C³ = … = C¹ ° ° = 0, the expansion simplifies to : 
A¹ ° ° = ¹ ° °C 0I + ¹ ° °C 1C 
A¹ ° ° = 1 · I + 100 · C 
A¹ ° ° = I + 100C 
Find Matrix B : 
The problem states that A¹ ° ° = 100B + I. Comparing this to our result: 
100B + I = 100C + I 
100B = 100C 
B = C 
Calculate the Sum of Elements of B¹ ° ° : 
Since B = C and we know C² = 0, then:   B² = 0 
Consequently, B¹ ° ° = 0 (the zero matrix). 
 
The sum of all the elements of B
100
 is 0. 
 
Q2: The number of 3 × 2 matrices A, which can be formed using the elements of the set 
{-2, -1, 0, 1, 2} such that the sum of all the diagonal elements of A?A is 5, is 
Ans: 312 
Sol: 
 
sum of diagonal elements of A?A = 5 
a² + c² + e² + b² + d² + f² = 5 
possible values of squares is {0, 1, 4} 
Case 1 : one element is 4,1 element is 1 and four elements are 0 . 
one element is 4 = 2 choices {-2, 2} 
one element is 1 = 2 choices {-1, 1} 
4 element is zero = 1 choice {0} 
Selecting one element 4 out of 6 choice {a², b², c², d², e², f²}, and have 2 choice {-2, 2}, total = 
6C 1 × 2 = 12 
Selecting one element which is 1 from remaining 5 element = 5C 1, and have two choice {-1, 1}, 
total = 5C 1 × 2 = 10 
Selecting 4 element which is zero from remaining 4 elements = 4C 4 = 1 total matrices for case 1 
= 12 × 10 × 1 = 120 
Case 2 : five elements are 1 and one element is 0. 
squares a², b², c², d², e², f² ? {1, 1, 1, 1, 1, 0} 
selection of 5 one from a², b², c², d², e², f² = 6C 5 
and for 1 there are two choice for each elements a, b, c, d, e, f = {-1, 1}, total = 2 5 
for 0 only 1 choices 
total matrices of case 2 = 6C 5 × 2 5 = 192 
total number of such matrices = Case 1 + Case 2 
= 120 + 192 = 312 
 
Q3: Let  and B be a matrix such that B (I - A) = I + A. Then 
the sum of the diagonal elements of B
T
B is equal to 
Ans: 3 
Sol: Observing properties of Matrix A 
 
Rearranging the given equation 
B(I - A) = I + A 
B = (I + A)(I - A) ?¹ 
Finding B? 
B? = [(I + A)(I - A) ?¹]? 
B? = ((I - A) ?¹)? (I + A)? 
B? = (I - A?) ?¹ (I + A?) 
Since A? = -A : 
B? = (I + A) ?¹(I - A) 
B?B = [(I + A) ?¹(I - A)] · [(I + A)(I - A) ?¹] 
B?B = (I + A) ?¹(I + A)(I - A)(I - A) ?¹ 
B?B = I · I = I 
 
the sum of diagonal elements = 1 + 1 + 1 = 3  
The sum of the diagonal elements of B
T
B is 3. 
 
 
Q4: Let |A| = 6, where A is a 3 × 3 matrix. If |adj (adj (A² · adj(2 A)))| = 2? · 3n, m, n ? N, 
then m + n is equal to____ . 
Ans: 62 
Sol: 
 
 
Page 5


 
JEE Main Previous Year Questions (2021-2026): 
Matrices and Determinants 
 
(January 2026) 
 
Q1: Let A = and B be two matrices such that A¹ ° ° = 100B + I. Then the sum of 
all the elements of B¹ ° ° is 
Ans: 0 
Sol: 
Decompose Matrix A  
We can write matrix A as the sum of the Identity matrix I and a residual matrix C: 
 
Since C² = 0, C is a nilpotent matrix. Consequently, Cn = 0 for all n = 2. 
Find A¹ ° ° using Binomial Expansion : 
A¹ ° ° = (I + C)¹ ° ° = ¹ ° °C 0I¹ ° ° + ¹ ° °C 1I ? ?C + ¹ ° °C 2I ? 8C² + … + ¹ ° °C 1 0 0C¹ ° ° 
Since C² = C³ = … = C¹ ° ° = 0, the expansion simplifies to : 
A¹ ° ° = ¹ ° °C 0I + ¹ ° °C 1C 
A¹ ° ° = 1 · I + 100 · C 
A¹ ° ° = I + 100C 
Find Matrix B : 
The problem states that A¹ ° ° = 100B + I. Comparing this to our result: 
100B + I = 100C + I 
100B = 100C 
B = C 
Calculate the Sum of Elements of B¹ ° ° : 
Since B = C and we know C² = 0, then:   B² = 0 
Consequently, B¹ ° ° = 0 (the zero matrix). 
 
The sum of all the elements of B
100
 is 0. 
 
Q2: The number of 3 × 2 matrices A, which can be formed using the elements of the set 
{-2, -1, 0, 1, 2} such that the sum of all the diagonal elements of A?A is 5, is 
Ans: 312 
Sol: 
 
sum of diagonal elements of A?A = 5 
a² + c² + e² + b² + d² + f² = 5 
possible values of squares is {0, 1, 4} 
Case 1 : one element is 4,1 element is 1 and four elements are 0 . 
one element is 4 = 2 choices {-2, 2} 
one element is 1 = 2 choices {-1, 1} 
4 element is zero = 1 choice {0} 
Selecting one element 4 out of 6 choice {a², b², c², d², e², f²}, and have 2 choice {-2, 2}, total = 
6C 1 × 2 = 12 
Selecting one element which is 1 from remaining 5 element = 5C 1, and have two choice {-1, 1}, 
total = 5C 1 × 2 = 10 
Selecting 4 element which is zero from remaining 4 elements = 4C 4 = 1 total matrices for case 1 
= 12 × 10 × 1 = 120 
Case 2 : five elements are 1 and one element is 0. 
squares a², b², c², d², e², f² ? {1, 1, 1, 1, 1, 0} 
selection of 5 one from a², b², c², d², e², f² = 6C 5 
and for 1 there are two choice for each elements a, b, c, d, e, f = {-1, 1}, total = 2 5 
for 0 only 1 choices 
total matrices of case 2 = 6C 5 × 2 5 = 192 
total number of such matrices = Case 1 + Case 2 
= 120 + 192 = 312 
 
Q3: Let  and B be a matrix such that B (I - A) = I + A. Then 
the sum of the diagonal elements of B
T
B is equal to 
Ans: 3 
Sol: Observing properties of Matrix A 
 
Rearranging the given equation 
B(I - A) = I + A 
B = (I + A)(I - A) ?¹ 
Finding B? 
B? = [(I + A)(I - A) ?¹]? 
B? = ((I - A) ?¹)? (I + A)? 
B? = (I - A?) ?¹ (I + A?) 
Since A? = -A : 
B? = (I + A) ?¹(I - A) 
B?B = [(I + A) ?¹(I - A)] · [(I + A)(I - A) ?¹] 
B?B = (I + A) ?¹(I + A)(I - A)(I - A) ?¹ 
B?B = I · I = I 
 
the sum of diagonal elements = 1 + 1 + 1 = 3  
The sum of the diagonal elements of B
T
B is 3. 
 
 
Q4: Let |A| = 6, where A is a 3 × 3 matrix. If |adj (adj (A² · adj(2 A)))| = 2? · 3n, m, n ? N, 
then m + n is equal to____ . 
Ans: 62 
Sol: 
 
 
Q5: Let A be a 3 × 3 matrix such that A + A
T
 = O. If 
 and det(adj(2 adj(A + I))) = (2)
a
 · (3)
ß
 
· (11)
?
 , a, ß, ? are non-negative integers, then a + ß + ? is equal to 
Ans: 18 
Sol: 
 
 
det(adj(2 adj(A + I))) = 2 6 det(adj(adj(A + I))) = 2 6(44) 4 = 2¹ 4 × 11 4 
 
Q6: For some a, ß ? R, let A =  and B =  be such that A² - 4A + 2I = 
B² - 3B + I = O. Then (det (adj (A³ - B³)))² is equal to ____ . 
Ans: 225