Matrices and Determinants — practice questions

143 curated practice questions from Matrices and Determinants (Mathematics JEE Main, NTA Exams) — 18 hard, 102 easy, 23 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• If S is the set of distinct values of b for which the following system of linear equations x+y+z=1 x+a y+z=1 a x+b…
• If A = [ array lll 0 & -1 & 2 2 & -2 & 0 array ], B = [ array ll 0 & 1 1 & 0 1 & 1 array ] and M = AB, then M –1 is…
• For which of the following ordered pairs ( , ) , the system of linear equations array l x+2 y+3 z=1 3 x+4 y+5 z= 4 x+4…
• The system of equations II x + y + z = II - 1 x + II y + z = II - 1 x + y + II z = II - 1 has no solution, if II is
• If A= [ array ll 1 & 0 1 & 1 array ] then A –n is equal to -
• If A= [ array cc 2 & -1 -7 & 4 array ] and B= [ array ll 4 & 1 7 & 2 array ] then B T A T is equal to –
• Given below are two statements. STATEMENT - 1: For a singular square matrix A, AB = AC ⇒ B = C STATEMENT - 2: If |A| =…
• If the trivial solution is the only solution of the system of equations x – ky + z = 0 kx + 3y – kz = 0 3x + y – z = 0…
• If A= [ array cc -1 & 7 2 & 3 array ] , then skew–symmetric part of A is –
• Consider the system of equation x+y+z=6, x+2 y+3 z=10 and x + 2 y + z= STATEMENT - 1: If the system has infinite number…
• If a + b + c = 0, one root of : | array ccc a-x & c & b c & b-x & a b & a & c-x array |=0 is
• Let A be any 3 3 invertible matrix. Then which one of the following is not always true?
• Find the inverse of the matrix A= [ array rrr 3 & -1 & -2 2 & 0 & -1 3 & -5 & 0 array ]
• The value of | array ccc 1 & & ^ 2 & ^ 2 & 1 ^ 2 & 1 & array |, ω being a cube root of unity, is
• If the system of linear equations array l x+y+z=5 x+z y+2 z=6 array x+3 y+ z= , ( , R), has infinitely many solutions…
• If A and B are square matrices of order n x n such that A^ 2 -B^ 2 =(A-B)(A+B) , then which of the following will be…
• If A + B = [ array ll 7 & 4 8 & 9 array ] and A-B= [ array ll 1 & 2 0 & 3 array ] then the value of A is-
• If the system of linear equations array l 2 x+2 y+3 z=0 3 x-y+5 z=b x-3 y+2 z=c array where a, b, c are non zero real…
• Consider the system of linear equations: x 1 + 2x 2 + x 3 = 3, 2x 1 + 3x 2 + x 3 = 3, 3x 1 + 5x 2 + 2x 3 = 1. The…
• Solution of x + 3y – 2z = 0 2x – y + 4z = 0 x – 11y + 14z = 0 is
• Inverse of the matrix [ array cc 1 & -2 3 & 4 array ] is:
• If A = [ array cc 2 & 3 5 & -2 array ], then 19 A –1 is equal to
• Inverse matrix of [ array cc 2 & -3 -4 & 2 array ] is -
• Let A and B be two invertible matrices of order 3 3 . If det(ABA T ) = 8 and det(AB -1 ) = 8, then det(BA -1 B T ) is…
• If A = [ array cc -1 & 2 3 & -4 array ] then element a 21 of A 2 is -
• If A= [ array ccc 2 & -2 & -4 -1 & 3 & 4 1 & -2 & x array ] is an idempotent matrix then x is :
• If A^ k =0 (null matrix) for some positive integral value of k, then I+ A + A ^ 2 + + A ^ k -1 is equal to
• If A = [ array ll 1 & 0 1 & 1 array ] and I = [ array ll 1 & 0 0 & 1 array ] , then which one of the following holds…
• The following system of linear equations 7x + 6y - 2z = 0 3x + 4y + 2z = 0 x - 2y - 6z = 0 has
• If A is an invertible matrix of order n, then the determinant of Adj. A =
• For two 3 3 matrices A and B, let A+B=2 B^ and 3 A+2 B=r_ 3 , where B ^ T is the transpose of B and I 3 is 3 3 identify…
• Let A be a 2 2 matrix Statement - 1: adj (adj A) = A Statement - 2: |adj A| = |A|
• If P= | array lll 2 & 1 & 0 3 & 1 & 2 5 & 2 & 3 array | , then | array lll 2 & 2 & 0 9 & 6 & 6 5 & 4 & 3 array | is…
• [ array cc 1 & - / 2 / 2 & 1 array ] [ array cc 1 & / 2 - / 2 & 1 array ]^ -1 is equal to
• If A= [ array ll 1 & 1 1 & 1 array ] . then A ^ 1 is equal to
• If A, B are two square matrices such that AB = A and BA = B, then
• The number of real values of for which the system of linear equations array l 2 x+4 y- z=0 4 x+ y+2 z=0 x+2 y+2 z=0…
• If the equations z x-y+z z=z x-2 y+z=-4 x+y+ z=4 have no solution, then the value of will be
• If A= [ array ccc 1 & -3 & 2 2 & k & 5 4 & 2 & 1 array ] is a singular matrix, then k is equal to
• If the system of equations x - ky - z = 0 kx - y - z = 0 x + y - z = 0 has a non-zero solution, then possible values of…
• Consider the system of linear equations x 1 + 2x 2 +x 3 = 3 2x 1 + 3x 2 + x 3 = 3 3x 1 + 5x 2 + 2x 3 = 1 The system has
• If the system of linear equations array l 2 x+2 a y+a z=0 2 x+3 b y+b z=0 2 x+4 c y+c z=0 array where a, b, c R are…
• STATEMENT - 1: A= [ array lll 4 & 0 & 4 2 & 3 & 2 1 & 2 & 1 array ], B= [ array lll 1 & 3 & 3 1 & 4 & 3 1 & 3 & 4 array…
• A 0.21 H inductor and a 12Ω resistance are connected in series to a 220 V, 50 Hz ac source. The phase angle between the…
• Given below are two statements. STATEMENT - 1: The inverse of the matrix A= [a_ i j ]_ n n where a _ ij =0, i j is E =…
• Let A be a square matrix of order n x n and K be a scalar. Then |KA| equals
• If A= [ array cc 4 & x +3 2 x -3 & x +1 array ] is a 2 2 symmetric matrix, then x is equal to
• If the system of equations 3x - 2y + z = 0 x - 14y + 15z = 0 x + 2y + 3z = 0 have a non-trivial solution, then the…
• Let A= ( array c - array ),( R) such that A^ 12 = ( array cc 0 & -1 1 & 0 array ) . Then the value of α is:
• The system of linear equations x + y + z = 2 2x + y - z = 3 3x + 2y + kz = 4 has a unique solution if
• If a, b, c are non-zero real numbers and if the system of equations (a - 1) x = y + z, (b - 1) y = z + x, (c -1)z = x +…
• Let A= ( array ccr 1 & -1 & 1 2 & 1 & -3 1 & 1 & 1 array ) and 108= ( array lcc 4 & 2 & 2 -5 & 0 & x 1 & -2 & 3 array )…
• The greatest value of c R for which the system of linear equations array l x-c y-c z=0 x-y+c z=0 x+c y-z=0 array has a…
• The root of the equation [ array lll x & 1 & 2 array ] [ array ccc 0 & 1 & 1 1 & 0 & 1 1 & 1 & 0 array ] [ array c x -1…
• If [1 x 2] [ array lll 2 & 3 & 1 0 & 4 & 2 0 & 3 & 2 array ] [ array c x 1 -1 array ] = O, then the value of x is
• If A= [ array ll 1 & 1 0 & 1 array ] and B= [ array cc 3 2 & 1 2 - 1 2 & 3 2 array ] then (B B^ A )^ 5 is equal to
• Given below are two statements. STATEMENT - 1: If b c+q r=c a+r p=a b+p q=-1 then | array ccc a p & a & p b q & b & q c…
• The number of values of k, for which the system of equations: array l (k+1) x+8 y=4 k k x+(k+3) y=3 k-1 array has no…
• STATEMENT - 1: | adj ( adj ( adj A))|= .|A|^ [n-1 |^ 3 , where n is order of matrix A. STATEMENT - 2: | adj A |=| A |^ n
• If the points P [x_ 1 , Y_ 1 ] , Q [x_ 2 , y_ 2 ] and R [x_ 3 , y_ 1 ] are collinear, then the rank of the matrix […
• The order of [xyz] [ array lll a & ~h & ~g ~h & ~b & f ~g & f & c array ] [ array l x y z array ] is
• If A= [ array lll 1 & 2 & 3 0 & 3 & 1 2 & 1 & 2 array ] , then A (adj A) equals -
• Given below are two statements. STATEMENT - 1: If a, b, c, d are real numbers and A= [ array ll a & b c & d array ] and…
• If A and B are 3 x 3 matrices such that AB = B and BA = A, then :
• If A= [ array ll 1 & 2 3 & 0 array ] and B= [ array ll 3 & 4 1 & 6 array ] then (AB) T equals -
• With 1, , 2 as cube roots of unity, inverse of which of the following matrices exist?
• The rank of matrix A= [ array rrrr 4 & 4 & 0 & 3 -2 & 3 & -1 & 5 1 & 4 & 8 & 7 array ] is :
• If E( )= [ array cc & - & array ] , then value of E (α) . E (β) is -
• If A= [ array cc 2 & -3 -4 & 1 array ] , then adj (3 ~A ^ 2 +12 ~A ) is equal to
• | array lll 11 & 2 & 3 2 & 3 & 4 3 & 4 & 5 array | is equal to
• If the system of linear equations x+2 a y+a z=0, x+3 b y+b z=0 and x+4 c y+c z=0 has a non-zero solution, then a, b, c
• If A= [ array cc p & q -q & p array ], B= [ array cc r & s -s & r array ] then
• The system of equations; x + y + z = 6, x + 2y + 3z = 10, x + 2y + z = has no solution for :
• If A= [ array ll a & b b & a array ] and A^ 2 = [ array ll x & & x array ] , then
• The system of linear equations array l x+ y-z=0 x-y-z=0 x+y- z=0 array has a non-trivial solution for:
• The value of h and m for which the system of equations x + y + z = 6 x + 2y + 3z = 10 x + 2y + hz = m has no solution…
• If the system of equations 2 x+3 y-z=0, x+k y-2 z=0 and z x-y+z=0 has a non-trivial solution (x, y, z) then x y + y z +…
• The determinant | array lll b _ 1 + c _ 1 & c _ 1 + a _ 1 & a _ 1 + b _ 1 ~b _ 2 + c _ 2 & c _ 2 + a _ 2 & a _ 2 + b _…
• The rank of matrix, A= [ array rrrr 3 & 3 & -1 & -1 1 & -1 & -2 & -4 3 & 1 & 3 & -2 6 & 3 & 0 & -7 array ] is :
• Matrix [ array ccc & -1 & 4 -3 & 0 & 1 -1 & 1 & 2 array ] is not invertible if -
• Let A and B be two matrices, then in general which of the following is true?
• Let A= [ array ccc x^ 2 & 6 & 8 3 & y^ 2 & 9 4 & 5 & z^ 2 array ], B= [ array ccc 2 x & 3 & 5 2 & 2 y & 6 1 & 4 & 2 z…
• If α, β & γ are the roots of the equation x 3 + px + q = 0 then the value of the determinant | array lll & & & & & &…
• Let A= ( array rrr 0 & 0 & -1 0 & -1 & 0 -1 & 0 & 0 array ) , the only correct statement about the matrix A is
• Consider, the system of equations k x+y+z=1 , x+k y+z=k and x+y+k z=k^ 2 . STATEMENT - 1 : System of equations has…
• If the system of equations x + y + z = 6, x + 2y + 3z = 10 and x + 2y + λ z = µ has no solution, then the values of λ…
• = | array ccc 18 & 40 & 89 40 & 89 & 198 89 & 198 & 440 array | is equal to
• If A = [ array ll 0 & 1 1 & 0 array ] , the A 4 =
• If A, B, C are matrices of order 1 × 3, 3 × 3 and 3 × 1 respectively, the order of ABC will be -
• If A= ( array ll 2 & 2 9 & 4 array ) and I= ( array ll 1 & 0 0 & 1 array ) then 10 ~A ^ -1 is equal to:
• If the matrix [ array ll a & b c & d array ] is commutative with the matrix [ array ll 1 & 1 0 & 1 array ] . then
• STATEMENT - 1: Every two-rowed real orthogonal matrix is of any one of the forms ( array cc & - & array ) or ( array cc…
• If A= [ array ccc 1 & -1 & 1 0 & 2 & -3 2 & 1 & 0 array ] , E= adj (A) , and C=5 A , then adj B |C| :
• If A is an 3 3 non-singular matrix such that A^ =A^ A and B=A^ -1 A^ , then BB' equals :
• Compute the adjoint of the matrix A= [ array lll 1 & 4 & 5 3 & 2 & 6 0 & 1 & 0 array ]
• Consider the system of equations a 1 x + b 1 y + c 1 z = 0, a 2 x + b 2 y + c 2 z = 0, a 3 x + b 3 y + c 3 z = 0 if |…
• For how many value(s) of x in the closed interval [-4,-1] , is the matrix [ array ccc 3 & -1+ x & 2 3 & -1 & x +2 x + 3…
• Given matrices A = [ array ll 1 & 2 3 & 4 array ] and [ array ll 5 & 6 7 & 8 array ], which of the following statements…
• If A and B are square matrices of same order, then which of the following is skew–symmetric –
• If . array ccc a-b-c & 2 a & 2 a 2 b & b-c-a & 2 b 2 c & 2 c & c-a-b array =(a+b+c)(x+a+b+c)^ 2 , x 0 and a+b+c 0 …
• The number of values of k, for which the system of equations (k + 1) x + 8y = 4k kx + (k + 3) y = 3k – 1 has no…
• If A= [ array cc 1 & -3 2 & k array ] and A^ 2 -4 A+10=A , then k is equal to
• If i = t 2 for 0 < t < T, then r.m.s. value of current is :
• If X and Y two matrices are such that X-Y= [ array cc 3 & 2 -1 & 0 array ] and X+Y= [ array cc 1 & -2 3 & 4 array ]…
• If A = [ array cc 1 & 2 3 & -5 array ], B = [ array ll 1 & 0 0 & 2 array ] and X is a matrix such that A = BX, then X…
• The number of values of θ (0, π) for which the system of linear equations array l x+3 y+7 z=0 -x+4 y+7 z=0 ( 38) x+(…
• The set of all values of for which the system of linear equations array l x-2 y-2 z= x x+2 y+z= y -x-y= z array has a…
• The number of 3 x 3 non-singular matrices, with four entries as 1 and all other entries as 0, is
• If A, B, C are non singular n x n matrices, then (A B C)^ -1 =
• A skew symmetric matrix S satisfies the relation 5^ 2 +I=0 , where I is a unit matrix, then SS' is equal to
• If a, b and c be positive integers and x=c y+b z, y=a z+c x, z=b x+a y , where x, y and z are not all zero. then number…
• A matrix A = [a i j ] of order 2 × 3 whose elements are such that a i j = i + j is -
• If A, B are two non-singular square matrices of the same order, then adj (AB) is equal to
• If A is a 3 x 3 skew - symmetric matrix then |A| is given by
• [ array ll -6 & 5 -7 & 6 array ]^ -1 =
• If the system of equations array l x+y+z=5 x+2 y+3 z=9 x+3 y+z z=3 array has infinitely many solutions, then β - α…
• Let P = [ array lll 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 array ] and Q = [q ij ] be two 3 3 matrices such that Q - P 5 = I 3 …
• If A= [ array lll 1 & 2 & 3 5 & 0 & 4 2 & 6 & 7 array ] then adj A is equal to -
• The value of a for which the system of equations x + y + z = 0 ax + (a + 1)y + (a + 2)z = 0 a 3 x + (a + 1) 3 y + (a +…
• If A= [ array cc 1 & 2 3 & -5 array ] . then A -1 is equal to
• If A = [ a ij ] is a skew-symmetric matrix of order n, then a ii =
• If | array ccc r & 2 r-1 & 3 r-2 n 2 & n-1 & a 1 2 n(n-1) & (n-1)^ 2 & 1 2 (n-1)(3 n-4) array |, then the value of _ r…
• Find the Adjoint of matrix A = [ array ccc 1 & 1 & 1 2 & 1 & -3 -1 & 2 & 3 array ]
• The system of equations kx + y + z = 1 x + ky + z = k x + y + kz = k 2 has no solution if k equals
• An ordered pair (α, β) for which the system of linear equations array l (1+a) x+ y+z=2 a x+(1+ ) y+z=3 a x+ y+2 z=2…
• The angular frequency of a.c. at which a coil of inductance 1mH has a reactance of 1 Ω is
• If every element of a third order determinant of value ∆ is multiplied by 5, then the value of new determinant is
• If [ array ll 1 & 2 2 & 1 array ] [ array l x y array ]= [ array l 5 4 array ] then
• Matrix [ array ccc 0 & 5 & -7 -5 & 0 & 11 7 & -11 & 0 array ] is a-
• The system of linear equations x + y +z = 2, 2x + y – z = 3, 3x + 2y + kz = 4 has a unique solution if
• Given a, b, c are in A.P. Then determinant | array lll x+1 & x+2 & x+a x+2 & x+3 & x+b x+3 & x+4 & x+c array | in its…
• If A= [ array lll 1 & 3 & 5 3 & 5 & 1 5 & 1 & 3 array ] , then adj. A is equal to -
• If the system of homogeneous equation x+a y=0, a z+y=0 and a x+z=0 has infinite number of solutions, then the value of…
• = | array ccc 0 & p-q & a-b q-p & 0 & x-y b-a & y-x & 0 array | is equal to
• If A and B are invertible matrices of the order n, then (AB) –1 is equal to
• The system of simultaneous equations k x+2 y-z=1 [k-1] y-2 z=z (k+2) z=3 have a unique solution if k equals
• With the help of matrices, the solution of the equations 3x + y + 2z = 3, 2x – 3y – z = – 3, x + 2y + z = 4 is given by
• If A= [ array ll 0 & 1 0 & 0 array ] and a and b are arbitrary constants then (aI + bA) 2 =
• STATEMENT - 1: = | array ccc m y+n z & m q+n r & m b+n c k z-m x & k r-m p & k c-m a -n x-k y & -n p-k q & -n a-k b…
• A generator produces a voltage that is given by V = 240 sin (120 t) volt, where t is in second. The frequency and…
• If the matrices A= [ array lll 2 & 1 & 3 4 & 1 & 0 array ] and B= [ array cc 1 & -1 0 & 2 5 & 0 array ] . then AB will…
• If A = [a ij ] is a square matrix of order n × n and k is a scalar, then | kA | =
• If the three linear equations x+4 a y+a z=0 , x + 3 b y + b z = 0 and x+2 c y+c z=0 have a non-trivial solution, then…

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