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Conjugate of a Matrix

If A=a11a12a1...

Question

If $A = \"|\"\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}\"|\"$ and C_ij is cofactor of a_ij in A, then value of |A| is given by

(a) a₁₁ C₃₁ + a₁₂C₃₂ + a₁₃ C₃₃
(b) a₁₁ C₁₁ + a₁₂ C₂₁ + a₁₃ C₃₁
(c) a₂₁ C₁₁ + a₂₂ C₁₂ + a₂₃ C₁₃
(d) a₁₁ C₁₁ + a₂₁ C₂₁ + a₁₃ C₃₁

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Solution

(d) a₁₁ C₁₁ + a₂₁ C₂₁ + a₁₃ C₃₁

Properties of determinants state that if A is a square matrix of the order n, then Det (A) is the sum of products of elements of a row (or a column) with the corresponding cofactor of that element.

$|A| = a_{11} C_{11} + a_{21} C_{21} + a_{31'} C_{31} [Calculating along C_{1}]$

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Similar questions

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 0 2 & 1 & 5 1 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

a_{11} A_{21} + a_{12} A_{22} + a_{13} A_{23} =

⎡ ⎢ ⎣ \begin{matrix} cos θ - sin θ 0 \end{matrix} \begin{matrix} sin θ cos θ 0 \end{matrix} \begin{matrix} 001 \end{matrix} ⎤ ⎥ ⎦

then the value of

a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} =

A_{11}, A_{12} A_{13}

are cofactors of

a_{11}, a_{12}, a_{13}

A = [a_{i j}]

4 \times 4

c_{i j}

is the co-factor of the element

a_{i j}

| A |

, then the expression

a_{11} c_{11} + a_{12} c_{12} + a_{13} c_{13} + a_{14} c_{14}

x \in R

S = 1 - C_{1} \frac{1 + x}{1 + n x} + C_{2} \frac{1 + 2 x}{{(1 + n x)}^{2}} - C_{3} \frac{1 + 3 x}{{(1 + n x)}^{3}} + . . . . . u p t o (n + 1) t e r m s

S

A

2 \times 2

matrices given by

A = [a_{i j}]

a_{i j} = {0, 1, 2, 3, 4}

a_{11} + a_{12} + a_{21} + a_{22} = 4

Find the number of matrices

A

such that the trace of

A

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