The following matrix is HERMATIAN Matrix begin bmatrix 1-i-5-i- 5 - 0 1 5 - 0 - 3end bmatrix

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Standard XII

Mathematics

Hermitian Matrix

The following...

Question

The following matrix is HERMATIAN Matrix $⎡ ⎢ ⎣ \begin{matrix} 1 & - i & 5 i & 5 & 0 5 & 0 & 3 \end{matrix} ⎤ ⎥ ⎦$

True

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False

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Solution

The correct option is A
True

We already saw that a Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate Transpose.

Here in option a : the transpose will be $⎡ ⎢ ⎣ \begin{matrix} 1 & i & 5 - i & 5 & 0 5 & 0 & 3 \end{matrix} ⎤ ⎥ ⎦$

the conjugate of this will be $⎡ ⎢ ⎣ \begin{matrix} 1 & - i & 5 i & 5 & 0 5 & 0 & 3 \end{matrix} ⎤ ⎥ ⎦$ which is same as original matrix

Therefore it is a Hermitian matrix.

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The following matrix is HERMATIAN Matrix $⎡ ⎢ ⎣ \begin{matrix} 1 & - i & 5 i & 5 & 0 5 & 0 & 3 \end{matrix} ⎤ ⎥ ⎦$

Q. Lets

A = [\begin{matrix} 0 & 5 - 5 & 0 \end{matrix}]

be a skew symmetric matrix and

I + A

is non singular, then the matrix

B = (I - A) (I + A)^{- 1}

Find the value of $x, y$ and $z$ from the following equation:
(i) $[\begin{matrix} 4 & 3 x & 5 \end{matrix}] = [\begin{matrix} y & z 1 & 5 \end{matrix}]$

(ii) $[\begin{matrix} x + y & 2 5 + z & x y \end{matrix}] = [\begin{matrix} 6 & 2 5 & 8 \end{matrix}]$

(iii) $⎡ ⎢ ⎣ \begin{matrix} x + y + z x + z y + z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 957 \end{matrix} ⎤ ⎥ ⎦$

Q. Show that

(i)

[\begin{matrix} 5 & - 1 6 & 7 \end{matrix}] [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] \neq [\begin{matrix} 2 & 1 3 & 4 \end{matrix}] [\begin{matrix} 5 & - 1 6 & 7 \end{matrix}]

(i i)

⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} \begin{matrix} 211 \end{matrix} \begin{matrix} 300 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} - 1 02 \end{matrix} \begin{matrix} 1 - 1 3 \end{matrix} \begin{matrix} 014 \end{matrix} ⎤ ⎥ ⎦ \neq ⎡ ⎢ ⎣ \begin{matrix} - 1 02 \end{matrix} \begin{matrix} 1 - 1 3 \end{matrix} \begin{matrix} 014 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} \begin{matrix} 211 \end{matrix} \begin{matrix} 300 \end{matrix} ⎤ ⎥ ⎦

Q. If

A = [\begin{matrix} 2 & - 5 0 & 1 \end{matrix}]

and

I = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

then find the matrix

X

such that

4 A - 2 X + I = 0

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The following matrix is HERMATIAN Matrix ⎡⎢⎣1−i5i50503⎤⎥⎦

The following matrix is HERMATIAN Matrix $⎡ ⎢ ⎣ \begin{matrix} 1 & - i & 5 i & 5 & 0 5 & 0 & 3 \end{matrix} ⎤ ⎥ ⎦$