Null Matrix

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 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.

10th Grade Math

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