Question

a)  Determine whether the matrix is singular.

Answer 1

A square matrix 'A' is called singular if it is not invertible. i.e. We can not find a matrix 'B' such that AB=I=BA. where I is the identity matrix.

The following are some conditions that is equivalent to being singular.

• The determinant of the matrix is ZERO.
• Row/column vectors are linearly DEPENDENT. i.e. some rows/columns are linear combination of other rows/columns.
• If the matrix is of order nxn then the rank of the matrix is STRICTLY LESS THAN n.
• 0 is an eigen value of the given matrix. ( For a square matrix A, a scalar p is said to be an eigen value if Av=pV for some non zero vector V).

Example :-

NILPOTENT MATRICES (Aⁿ=O for some n) , ODD ORDER SKEW SYMMETRIC MATRICES (A=-A^T) are always singular.

(No particular matrix is given in the question. The above are some ways for checking wheather a matrix is singular or not.)