Question

Can we say that a zero matrix is invertible?

Answer 1

Hint: A bit of knowledge of matrices like, what is a matrix, zero matrices, identity matrix, the inverse of a matrix is enough to solve this question and all formulas of the matrix should be clear. Firstly we will check the condition for the invertible matrix then we will check whether the zero matrices follow the condition to be invertible or not.

Complete step-by-step solution:
Matrix is a set of numbers, symbols, or expressions arranged in rows and columns to form a rectangular array. The numbers, symbols, or expressions are called the elements of the matrix.
A Matrix can be used to compactly write and work with multiple linear equations.
A zero matrix is just a matrix with all the elements inside the matrix as 0. It does not have to be a square matrix.
\[A=\left[ \begin{align}
  & \text{0 0 0} \\
 & \text{0 0 0} \\
 & \text{0 0 0} \\
\end{align} \right]\]
Here A is a zero matrix of \[3\times 3\] order, that is it has three rows and three columns and all the elements of the matrix are zero.
An Identity matrix is a square matrix of any order where the principal diagonal has elements as ones and other elements as zeros.
\[I=\left[ \begin{align}
  & \text{1 0 0} \\
 & \text{0 1 0} \\
 & \text{0 0 1} \\
\end{align} \right]\]
Here A is an identity matrix of \[3\times 3\]order.
An Invertible matrix is a type of square matrix containing real and complex numbers. Its main characteristics are that for an invertible matrix there is always another matrix which when multiplied to the first, will produce the identity matrix of the same dimensions as them.
\[A\cdot {{A}^{-1}}={{I}_{n}}\]
Now, the condition for a matrix to be invertible is that: The matrix is invertible if and only if its determinant is not zero.
Now, a zero matrix cannot be an invertible matrix because of the following reasons:
1) No matter which matrix you multiply to a zero matrix and no matter the order in which the multiplication occurs, the result of such matrix will always be a zero matrix.
That is \[{{0}_{n}}\cdot A={{0}_{n}}\]
2) Determinant of the zero matrices is 0.
\[\therefore \] A zero matrix cannot be an invertible matrix.


Note: Two matrices can be multiplied with each other even if they have different dimensions, as long as the number of columns in the first matrix is equal to the number of rows as the first matrix is equal to the number of rows in the second matrix. The multiplication of matrices is not commutative but the multiplication of matrices is associative.