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Question

If w is a complex cube root of unity, then the matrix A = $open square brackets table row 1 cell w to the power of 2 end exponent end cell w row cell w to the power of 2 end exponent end cell w 1 row w 1 cell w to the power of 2 end exponent end cell end table close square brackets$ is a-

singular matrix
non-singular matrix
skew symmetric matrix
None of these

The correct answer is: singular matrix

We have
|A|= $open square brackets table row 1 cell w to the power of 2 end exponent end cell w row cell w to the power of 2 end exponent end cell w 1 row w 1 cell w to the power of 2 end exponent end cell end table close square brackets$ = $open square brackets table row cell 1 plus w to the power of 2 end exponent plus w end cell cell w to the power of 2 end exponent end cell w row cell w to the power of 2 end exponent plus w plus 1 end cell w 1 row cell w plus 1 plus w to the power of 2 end exponent end cell 1 cell w to the power of 2 end exponent end cell end table close square brackets$ [Using $C subscript 1 end subscript rightwards arrow C subscript 1 end subscript plus C subscript 2 end subscript plus C subscript 3 end subscript$ ]
= $open square brackets table row 0 cell w to the power of 2 end exponent end cell w row 0 w 1 row 0 1 cell w to the power of 2 end exponent end cell end table close square brackets$ = 0
A is a singular matrix.

Related Questions to study

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parallel

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Statement I : $open square brackets table row 5 0 0 row 0 3 0 row 0 0 2 end table close square brackets$ is a diagonal matrix.
Statement II : A square matrix A = (aij) is a diagonal matrix if aij = 0 . $for all straight i times not equal to straight j$

Statement I : $open square brackets table row 5 0 0 row 0 3 0 row 0 0 2 end table close square brackets$ is a diagonal matrix.
Statement II : A square matrix A = (aij) is a diagonal matrix if aij = 0 . $for all straight i times not equal to straight j$

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity. Denote by tr(A), The sum of diagonal entries of A, Assume that A² = I.
Statement-I :If A $not equal to$ I and A $not equal to$ – I, then det A= – 1
Statement-II : If A $not equal to$ I and A $not equal to$ – I then tr(A) $not equal to$ 0.

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity. Denote by tr(A), The sum of diagonal entries of A, Assume that A² = I.
Statement-I :If A $not equal to$ I and A $not equal to$ – I, then det A= – 1
Statement-II : If A $not equal to$ I and A $not equal to$ – I then tr(A) $not equal to$ 0.

parallel

Suppose $A equals open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets$ , $B equals open square brackets table row 2 0 row 0 cell negative 2 end cell end table close square brackets$ let x be a 2×2 matrix such that $X to the power of straight prime$ AX = B
Statement-I : X is non singular & |x| = ±2
Statement-II : X is a singular matrix

Suppose $A equals open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets$ , $B equals open square brackets table row 2 0 row 0 cell negative 2 end cell end table close square brackets$ let x be a 2×2 matrix such that $X to the power of straight prime$ AX = B
Statement-I : X is non singular & |x| = ±2
Statement-II : X is a singular matrix

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Statement-I : If A & B are two 3×3 matrices such that AB = 0, then A = 0 or B = 0
Statement-II : If A, B & X are three 3×3 matrices such that AX = B, |A| $not equal to$ 0, then X = A^–1B

Statement-I : If A & B are two 3×3 matrices such that AB = 0, then A = 0 or B = 0
Statement-II : If A, B & X are three 3×3 matrices such that AX = B, |A| $not equal to$ 0, then X = A^–1B

parallel

Assertion (A): The inverse of the matrix $open square brackets table row 1 3 5 row 2 6 10 row 9 8 7 end table close square brackets$ does not exist.
Reason (R) : The matrix $open square brackets table row 1 3 5 row 2 6 10 row 9 8 7 end table close square brackets$ is singular. [ $because$ $open vertical bar table row 1 3 5 row 2 6 10 row 9 8 7 end table close vertical bar$ = 0, since R₂ = 2R₁]

Assertion (A): The inverse of the matrix $open square brackets table row 1 3 5 row 2 6 10 row 9 8 7 end table close square brackets$ does not exist.
Reason (R) : The matrix $open square brackets table row 1 3 5 row 2 6 10 row 9 8 7 end table close square brackets$ is singular. [ $because$ $open vertical bar table row 1 3 5 row 2 6 10 row 9 8 7 end table close vertical bar$ = 0, since R₂ = 2R₁]

Assertion (A): $open square brackets table row 5 0 0 row 0 3 0 row 0 0 9 end table close square brackets$ is a diagonal matrix
Reason (R) : A square matrix A = (a_ij) is a diagonal matrix if a_ij = 0 for all i $not equal to$ j.

Assertion (A): $open square brackets table row 5 0 0 row 0 3 0 row 0 0 9 end table close square brackets$ is a diagonal matrix
Reason (R) : A square matrix A = (a_ij) is a diagonal matrix if a_ij = 0 for all i $not equal to$ j.

Consider $open vertical bar table row cell a subscript 1 end subscript end cell cell b subscript 1 end subscript end cell cell c subscript 1 end subscript end cell row cell a subscript 2 end subscript end cell cell b subscript 2 end subscript end cell cell c subscript 2 end subscript end cell row cell a subscript 3 end subscript end cell cell b subscript 3 end subscript end cell cell c subscript 3 end subscript end cell end table close vertical bar$ = – 1, where a_i. a_j + b_i. b_j + c_i.c_j = $open square brackets table row cell 0 semicolon end cell cell i not equal to j end cell row cell 1 semicolon end cell cell i equals j end cell end table close$ and i, j = 1,2,3
Assertion(A) : The value of $open vertical bar table row cell a subscript 1 end subscript plus 1 end cell cell b subscript 1 end subscript end cell cell c subscript 1 end subscript end cell row cell a subscript 2 end subscript end cell cell b subscript 2 end subscript plus 1 end cell cell c subscript 2 end subscript end cell row cell a subscript 3 end subscript end cell cell b subscript 3 end subscript end cell cell c subscript 3 end subscript plus 1 end cell end table close vertical bar$ is equal to zero
Reason(R) : If A be square matrix of odd order such that AA^T = I, then | A + I | = 0

Consider $open vertical bar table row cell a subscript 1 end subscript end cell cell b subscript 1 end subscript end cell cell c subscript 1 end subscript end cell row cell a subscript 2 end subscript end cell cell b subscript 2 end subscript end cell cell c subscript 2 end subscript end cell row cell a subscript 3 end subscript end cell cell b subscript 3 end subscript end cell cell c subscript 3 end subscript end cell end table close vertical bar$ = – 1, where a_i. a_j + b_i. b_j + c_i.c_j = $open square brackets table row cell 0 semicolon end cell cell i not equal to j end cell row cell 1 semicolon end cell cell i equals j end cell end table close$ and i, j = 1,2,3
Assertion(A) : The value of $open vertical bar table row cell a subscript 1 end subscript plus 1 end cell cell b subscript 1 end subscript end cell cell c subscript 1 end subscript end cell row cell a subscript 2 end subscript end cell cell b subscript 2 end subscript plus 1 end cell cell c subscript 2 end subscript end cell row cell a subscript 3 end subscript end cell cell b subscript 3 end subscript end cell cell c subscript 3 end subscript plus 1 end cell end table close vertical bar$ is equal to zero
Reason(R) : If A be square matrix of odd order such that AA^T = I, then | A + I | = 0

parallel

Assertion(A) : The inverse of the matrix A = [Aij]_{n × n} where a_ij = 0, i $greater or equal than$ j is B = [aij^–1]_{n× n}
Reason(R): The inverse of singular matrix does not exist

Assertion(A) : The inverse of the matrix A = [Aij]_{n × n} where a_ij = 0, i $greater or equal than$ j is B = [aij^–1]_{n× n}
Reason(R): The inverse of singular matrix does not exist

Assertion : The product of two diagonal matrices of order 3 × 3 is also a diagonal matrix
Reason : matrix multiplicationis non commutative

Assertion : The product of two diagonal matrices of order 3 × 3 is also a diagonal matrix
Reason : matrix multiplicationis non commutative

Assertion : If A is a skew symmetric of order 3 then its determinant should be zero.
Reason : If A is square matrix then det A = det $straight A to the power of straight prime$ = det (– $straight A to the power of straight prime$ )

Assertion : If A is a skew symmetric of order 3 then its determinant should be zero.
Reason : If A is square matrix then det A = det $straight A to the power of straight prime$ = det (– $straight A to the power of straight prime$ )

parallel

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