Conjugate of a Matrix - Practice Questions & MCQ

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 . (If the element of a matrix is a + ib, then it is replaced by a - ib .)
 

Properties of the conjugate of a matrix:

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

i)

ii)  where A and B are conformable for matrix addition.

iii)   where A and B are conformable for multiplication.

iv) , where k is real or complex.

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

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.