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Conjugate of a Matrix - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Transpose conjugate of a matrix and properties is considered one the most difficult concept.

  • 11 Questions around this concept.

Solve by difficulty

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.

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.

 

Study it with Videos

Conjugate of a Matrix
Transpose conjugate of a matrix and properties

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