Conjugate of a Matrix MCQ - Practice Questions & Answers

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Transpose conjugate of a matrix and properties is considered one the most difficult concept.
11 Questions around this concept.

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EASY

Let A and B be two symmetric matrices of order 3 .

Statement -1 : $A(BA)$ and $(AB)A$ are symmetric matrices.

Statement -2 : $AB$ is symmetric matrix if matrix multiplication of A and B is commutative.

Statement-1 is true, Statement-2 is false.

Statement-1 is false, Statement-2 is true.

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 .

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

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Concepts Covered - 2

Conjugate of a Matrix

If a matrix A has a complex numbers as it’s elements, then the matrix obtained by replacing those complex number by their conjugates is called conjugate of the matrix A and it is denoted by $\\\mathrm{\overline{A}}$ . (If the element of a matrix is a + ib, then it is replaced by a - ib .)

$\\\mathrm{e.g.\;\; \\A = \begin{bmatrix} 2i &3+4i & 7\\ 3i & 9 & 4+5i\\ 4+5i & 4i & 3+7i \end{bmatrix}\; then, } \\\\\\\mathrm{\;\;\;\;\;\;\overline{A}=\begin{bmatrix} -2i &3-4i & 7\\ -3i & 9 & 4-5i\\ 4-5i & -4i & 3-7i \end{bmatrix}}$

Properties of the conjugate of a matrix:

If A and B are two matrices of the same order, then

i) $\\\mathrm{ \overline{(\overline{A})} = A}$

ii) $\\\mathrm{ \overline{(A + B)} = \overline{A}+\overline{B}}$ where A and B are conformable for matrix addition.

iii) $\\\mathrm{ \overline{(A\times B)} = \overline{A}\times\overline{B}}$ where A and B are conformable for multiplication.

iv) $\mathrm{\overline{(kA)}=\overline{k}\;\cdot\overline{A}}$ , where k is real or complex.

Transpose conjugate of a matrix and properties

Transpose conjugate of a matrix and properties:

The transpose of a conjugate matrix A is called the transposed conjugate of A and is denoted by A^?. The conjugate of the transpose of A is the same as the transpose of the conjugate of A

$\mathrm{i.e.\;\;A^\theta=(\overline A)'=\overline{(A')}}$

$\\\mathrm{A=\begin{bmatrix} 1+2i &3i &5+4i \\ 2i-1& 1-i &0 \\ 3+i&1+i &12 \end{bmatrix}}\\\\\\\mathrm{\overline{A}=\begin{bmatrix} 1-2i &-3i &5-4i \\ -2i-1& 1+i &0 \\ 3-i&1-i &12 \end{bmatrix}}\\\\\\\mathrm{\left (\overline{A} \right )'=\begin{bmatrix} 1-2i &-2i-1 &3-i \\ -3i & 1+i &1-i \\ 5-4i&0 &12 \end{bmatrix}}$

Properties of the transpose conjugate matrix:

If A and B are two matrices of the same order then

i) conjugate of a conjugate of matrix is the same as the original matrix itself,

In mathematical language (A^?)^? = A, which is quite obvious as we are reversing back the things which we did while taking conjugate at first time.

ii) (A + B)^? = A^? + B^?, this is obvious if a matrix is conformable, as addition is done element-wise.

Iii )(kA)^? = kA^?, since multiplication in a matrix are elementwise, hence this is also obvious, as all elements multiplied before conjugate will be amplified in the way as after taking conjugate.

iv) (AB)^? = B^?A^?, here A and B should be conformable for matrix multiplication.