If matrix A - - 1 amp-0 amp-1- 3 amp-4 amp-5- 0 amp-6 amp-7 -and its inverse is denoted by A-1 - - a-11 amp- a-12 amp- a-13- a-21 amp- a-22 amp- a-23- a-31 amp- a-32 amp- a-33 - then the value of a-23-21-20-2-51-52-5

The correct option is D 2/5|A| = 20 ∴ a_23 = Cofactor of 6/ 20 = 8/ 20 = 2/5 ...

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Inverse of a matrix

If matrix A =...

Question

If matrix $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 1 3 & 4 & 5 0 & 6 & 7 \end{matrix} ⎤ ⎥ ⎦$ and its inverse is denoted by $A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ⎤ ⎥ ⎦$ , then the value of $a_{23}$ =

\frac{21}{20}

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\frac{1}{5}

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- \frac{2}{5}

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\frac{2}{5}

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Solution

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A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 1 3 & 4 & 5 0 & 6 & 7 \end{matrix} ⎤ ⎥ ⎦

A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ⎤ ⎥ ⎦

a_{23} =

A = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & - 1 3 & 4 & 5 0 & 6 & 7 \end{matrix} ∣ ∣ ∣ ∣

A^{- 1} = ∣ ∣ ∣ ∣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ∣ ∣ ∣ ∣

a_{23}

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 1 3 & 4 & 5 0 & 6 & 7 \end{matrix} ⎤ ⎥ ⎦

A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ⎤ ⎥ ⎦

a_{23}

Δ_{0} = ⎡ ⎢ ⎣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ⎤ ⎥ ⎦

Δ_{0} \neq 0

Δ_{1}

Δ_{0}

Δ_{2}

Δ_{1}

Δ_{n}

Δ_{n - 1}

Δ_{n}

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \frac{- 2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦, B = [\begin{matrix} 1 & 0 i & 1 \end{matrix}], i = \sqrt{- 1}

Q = A^{T} B A,

A Q^{2021} A^{T}

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