
Can we say that a zero matrix is invertible?
Answer
422.7k+ views
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.
Here A is a zero matrix of 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.
Here A is an identity matrix of 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.
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
2) Determinant of the zero matrices is 0.
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.
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.
Here A is a zero matrix of
An Identity matrix is a square matrix of any order where the principal diagonal has elements as ones and other elements as zeros.
Here A is an identity matrix of
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.
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
2) Determinant of the zero matrices is 0.
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.
Latest Vedantu courses for you
Grade 9 | CBSE | SCHOOL | English
Vedantu 9 CBSE Pro Course - (2025-26)
School Full course for CBSE students
₹37,300 per year
Recently Updated Pages
Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Biology: Engaging Questions & Answers for Success

Master Class 12 Physics: Engaging Questions & Answers for Success

Master Class 12 English: Engaging Questions & Answers for Success

Trending doubts
Why should a magnesium ribbon be cleaned before burning class 12 chemistry CBSE

A renewable exhaustible natural resources is A Coal class 12 biology CBSE

Megasporangium is equivalent to a Embryo sac b Fruit class 12 biology CBSE

What is Zeises salt and ferrocene Explain with str class 12 chemistry CBSE

How to calculate power in series and parallel circ class 12 physics CBSE

Anal style is present in A Male cockroach B Female class 12 biology CBSE
