Null Matrix | Null or Zero Matrix|Zero Matrix|Problems on Null Matrix

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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 Om,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 A2 + 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 A2 = (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, A2 + 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, A2 + I = 0.

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10th Grade Math

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