If a square matrix A has two identical rows or columns , then det.A is :a)0b)-1c)1d)none of these.Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Answer

Explanation:

Let A be a square matrix of order n

If A has two identical rows or columns, then the determinant of A is 0.

This can be proved by using the property of determinants that if any two rows (or columns) of a matrix are identical, then its determinant is 0.

Let A have two identical rows. Without loss of generality, let the first two rows be identical.

Let B be the matrix obtained by deleting the first row of A.

Then A and B have the same determinant because multiplying the first row of A by (-1) and adding it to the second row gives B without changing the determinant.

Since the first two rows of A are identical, the determinant of A is the same as the determinant of B with an extra factor of 0.

Therefore, det(A) = 0 * det(B) = 0.

Similarly, if A has two identical columns, we can delete one of them and repeat the above argument.

Example:
Consider the following matrix A:
\begin{bmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
1 & 2 & 3 \
\end{bmatrix}
The first and third rows of A are identical, so det(A) = 0.

Answer

Explanation:

Let A be a square matrix of order n x n.

If A has two identical rows or columns, then the determinant of A is zero.

Proof:

Without loss of generality, let A have two identical rows, say ith and jth rows.

Then, if we interchange the ith and jth rows of A, we obtain a new matrix B, which is similar to A.

Now, det(B) = - det(A) (since interchanging two rows changes the sign of the determinant).

But, B has two identical rows. Therefore, det(B) = 0.

Therefore, - det(A) = 0, which implies det(A) = 0.

Hence, if A has two identical rows or columns, then det(A) = 0.

Example:

Consider the following matrix A, which has two identical rows:

A =
\begin{pmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
1 & 2 & 3 \
\end{pmatrix}

Interchanging the first and third rows of A, we obtain the matrix B:

B =
\begin{pmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
1 & 2 & 3 \
\end{pmatrix}

It is clear that B is similar to A, and det(B) = 0 since it has two identical rows.

Therefore, det(A) = - det(B) = 0.

Hence, the answer is (a) 0.

Answer

It is the property of determinants that a matrix has 2 identical rows or columns then determinant of that matrix will be zero

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