If each element of an m × n matrix be 0, the null element of

F, the matrix is said to be the null matrix or the zero matrix of order m × n and

it is denoted by O_{m,n}. It is also denoted by O, when no confusion

regarding its order arises.

Null or zero Matrix: Whether A is a rectangular or square

matrix, A – A is a matrix whose every element is zero. The matrix whose every

element is zero is called a null or zero matrix and it is denoted by 0.

Thus for A and 0 of the same order we have A + 0 = A

For example,

(begin{bmatrix} 0 & 0 end{bmatrix}) is a zero matrix of order 1 × 2.

(begin{bmatrix} 0\ 0 end{bmatrix}) is a zero or null matrix of order 2 × 1.

(begin{bmatrix} 0 & 0\ 0 & 0 end{bmatrix}) is a null matrix of order 2 × 2.

(begin{bmatrix} 5 & 6 & 4\ 1 & 0 & 9 end{bmatrix}) is a null matrix of order 2 × 3.

Problems on Null or zero matrix:

**1.** Find two nonzero matrices whose product is a zero matrix.

**Solution:**

Let A = (begin{bmatrix} 1 & 0\ 0 & 1

end{bmatrix}) and B = (begin{bmatrix} 0 & 0\ 0 & 1 end{bmatrix})

be two non-zero matrices.

But AB = (begin{bmatrix} 1 & 0\ 0 & 1

end{bmatrix}) (begin{bmatrix} 0 & 0\ 0 & 1 end{bmatrix}) = (begin{bmatrix}

0 & 0\ 0 & 0 end{bmatrix}) is a null matrix.

**2.** If A = (begin{bmatrix} 1 & 2\ -1 & -1

end{bmatrix}), show that A^{2} + I = 0.

(I and 0 being identity and null matrices of order 2).

**Solution:**

Given, A = (begin{bmatrix} 1 & 2\ -1 & -1

end{bmatrix})

Now A^{2 }= (begin{bmatrix} 1 & 2\ -1 & -1

end{bmatrix})(begin{bmatrix} 1 & 2\ -1 & -1 end{bmatrix}) = (begin{bmatrix}

-1 & 0\ 0 & -1 end{bmatrix})

Therefore, A^{2 }+ I = (begin{bmatrix} -1 & 0\

0 & -1 end{bmatrix}) + (begin{bmatrix} 1 & 0\ 0 & 1

end{bmatrix}) = (begin{bmatrix} 0 & 0\ 0 & 0 end{bmatrix})

Thus, A^{2} + I = 0.

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