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the first column of M is the transpose of the second row of Mthe second row of M is the transpose of the column of MM is a diagonal matrix with non-zero entries in the main diagonalthe product of entries in the main diagonal of M is not the square of an integer

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C::DSolution :

Let `M=[(a,c),(c,b)]" "("where "a, b, c, in I)` <br> (1) If the first column of M is the transpose of the second row of M, then <br> `[(a,c)]=[(c,b)]` <br> `:. A=b=c` <br> Thus, det. `(M) =ab-c^(2)=0` <br> Hence, `M` is not invertible. <br> (2) If the second row of `M` is the transpose of the first column of M, then <br> `[(c, b)]=[(a, c)]` <br> `:. a=b=c` <br> thus, det `(M)=ab-c^(2)=0` <br> Hence, `M` is not invertible. <br> (3) If `M=[(a,0),(0,b)]`, with `a, b, ne 0`, then <br> det. `(M)=ab ne 0` <br> Hence, `M` is invertible. <br> (4) If product of elements in main diagonal which `(ab)` is not perfect square, then <br> det. `(M)=ab-c^(2) ne 0` <br> Hence, `M` is invertiable.**Definitions; matrix representation; rows; column or general element**

**Row matrix and column matrix**

**Square matrix and diagonal matrix**

**Scalar matrix and identity matrix**

**Null matrix upper triangular matrix and lower triangular matrix**

**Symmetric matrix**

**skew symmetric matrix**

**Let `A` and `B` be symmetric matrices of same order. Then `A+B` is a symmetric matrix, `AB-BA` is a skew symmetric matrix and `AB+BA` is a symmetric matrix**

**Every matrix can be represented as a sum of symmetric and skew symmetric matrices**

**Singular matrix and Non-Singular Matrix**