Question

please help to answer below questions. Multiple answers can also be possible.

D Question 4 1 pts Let B-|b, b2 br 1 b br+1 -br) be a non-singular matrix. If column br is replaced by a' and that the resulting matrix is called Ba along with a-Σ-1 yibi. then state the necessary and suffcient condition for Ba to be non-singular. One of y i not necessarilyy.s, is zero is sufficient y ris non-zero is necessary ■ yr is non-zero is sufficient One ofy i, not necessarily y_r, is zero is necessary Previous Next . D Question 1 1 pts LetAmxn be a given matrix. Give an upper bound on the number of times the determinant function would be used to determine the rank of this matrix nCn+nCin-1nC2 for a case mCn nCm mCm+mCm-1)mC2 for a case Next Question 3 1pts Let A be an nx n matrix obeying the equation A2 A, the possible values of det(A) O 0or 1 neitherOnor 1 Previous Next

Answer 1

4. Since the matrix B is non singular so columns of B are linearly independent .

Now we have replaced b_r by  $\sum_{i=1}^{n} \gamma_{i} b_{i}$

So our required matrix is non-singular if it's columns are still independent so $\gamma_{r}$ is non zero , otherwise r-th column can be written as linear combination of others .

Hence ,

Answer : b , c .

1. Suppose   m $\geq$ n .

First we have to check all n × n block determinant , then all n-1 × n-1 block determinant ,......, all 2×2 determinant.

So number of times to determine determinant is ,

=

2 rl

Similarly if n$\geq$ m then number of times is ,

7m -1

Hence ,

Answer : a , d .

3 . Given ,

A² = A

Or , det(A²) = det(A)

Or , (detA)² = det(A)

Or , det(A) = 1 or 0

Hence ,

Answer : c .

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