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The determinant zero, if
Operating C_{3}→ C_{3} – C_{1} α – C_{2}, we get
⇒ (ac – b^{2}) (aα^{2} + 2bα + c) = 0
⇒ either ac – b^{2} = 0 or aα^{2} + 2b α + c = 0
⇒ either a, b, c are in G.P. or (x – α) is a factor of ax^{2} + 2bx + c
⇒ (b) and (e) are the correct answers.
[∴ C_{2} and C_{3} are identical]
⇒ x + iy = 0 ⇒ x = 0, y = 0
Let M and N be two 3 × 3 nonsingular skew symmetric matrices such that MN = NM. If PT denotes the transpose of P, then M^{2}N^{2} (M^{T}N)^{–1} (MN^{–1})^{T} is equal to
[As a skew symmetric matrix of order 3 cannot be non singular, therefore the data given in the question is inconsistent.]
We have M^{2}N^{2} (M^{T} N)^{–1} (MN^{–1})T = M^{2}N^{2}N^{–1} (M^{T})^{–1} (N^{–1})^{T}
M^{T}
= M^{2} N (M^{T})^{–1} (N^{–1})^{T} M^{T} = –M^{2}NM^{–1} N^{–1}M
(∵ M^{T} = –M, N^{T} = –N and (N^{–1})T = (N^{T})^{–1}
= – M (NM) (NM)^{–1} M (∵ MN = NM)
= – MM = –M^{2}
If the adjoint of a 3 x 3 matrix P is then the possible value(s) of the determinant of P is (are)
ANSWER : a,d
Solution : adj P = P^2 as (adj(P) = P^(n1))
adj P = 1(37) 4(67) +4(21) = 4
Hence, P = 2 or 2
For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct?
(a)(N' M N)' = (M N)'N = N'M 'N = N'M N or –N'M N According as M is symm. or skew symm. ∴correct
(b) (MN – NM)' = (MN)' – (NM)' = N'M' – M'N' = NM – MN = –(MN – NM)
∴ It is skew symm. Statement B is also correct.
(c)(MN)' = N'M' = NM ¹ MN
∴ Statement C is incorrect
(d) (adj M) (adj N) = adj (MN) is incorrect.
Let ω be a complex cube root of unity with ω ≠ 1 and P = [p_{ij}] be a n × n matrix with p_{ij} = ω^{i+j}. Then p^{2} ≠ 0, when n =
It shows P^{2} = 0 if n is a multiple of 3.
So for P^{2} ≠ 0, n should not be a multiple of 3 i.e. n can take values 55, 58, 56
Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if
where a, b, c are integers.
M is invertible if
∴ (a) is not correct.
If [bc] = [ab] ⇒ b = a = c ⇒ ac = b^{2}
∴ (b) is not correct.
∴ M is invertible.
(c) is correct
As ac ≠ (integer)2 ⇒ ac ≠ b^{2}
∴ (d) is correct.
Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N^{2} and M^{2} = N^{4}, then
Given MN = NM, M ≠ N^{2} and M^{2} = N^{4}.
Then M^{2} = N^{4} ⇒ (M + N^{2}) (M – N ^{2}) = 0
⇒ (i) M + N^{2} = 0 and M – N^{2} ≠ 0
(ii) M + N^{2} = 0 and M – N^{2} = 0
In each case M + N^{2} = 0
∴ M^{2} + MN^{2} = M M + N^{2} = 0
∴ (a) is correct and (c) is not correct.
Also we know if A = 0, then there can be many matrices U, such that AU = 0
∴ (M^{2} + MN^{2})U = 0 will be true for many values of U.
Hence (b) is correct.
Again if AX = 0 and A = 0, then X can be nonzero.
∴ (d) is not correct.
Which of the following values of a satisfy the equation
⇒ 2α^{2}(–2α) = –324α ⇒ α^{3} – 81α = 0 ⇒ α = 0, 9, –9
Let X and Y be two arbitrary, 3 × 3, nonzero, skewsymmetric matrices and Z be an arbitrary 3 × 3, non zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
X ' = –X, Y ' = –Y, Z ' = Z
(Y^{3}Z^{4} – Z^{4}Y^{3})' = (Z^{4})'(Y^{3})' – (Y^{3})'(Z^{4})'
= (Z')4(Y')3 – (Y')3(Z')4
= –Z^{4}Y^{3} + Y^{3}Z^{4} = Y^{3}Z^{4} – Z^{4}Y^{3}
∴ Symmetric matrix.
Similarly X^{44} + Y^{44} is symmetric matrix and X^{4}Z^{3} – Z^{3}X^{4} and X^{23} + Y^{23} are skew symmetric matrices.
Suppose Q = [qij] is a matrix such that PQ = kI, where and I is the identity matrix of order 3. then
Comparing the third elements of 2nd row on both sides, we get
Let Consider th e system of lin ear equations
ax + 2y = λ
3x – 2y = μ
Which of the following statement(s) is (are) correct?
ax + 2y = λ
3x – 2y = μ
For unique solution,
∴ (b) is the correct option.
For infinite many solutions and a = – 3
∴ (c) is the correct option.
⇒ system has no solution.
⇒ (d) is the correct option.
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