7. Given ) x+y=0 x-3y+4z=0 ,which one of the ff alternatives is/are solution(s)of the given system of linear equation? A. (} -1 1 1 ) B. (} 1 -1 -1 ) C. (} 0.5 -0.5 -0.5 ) D. All E. None 8.Given matrix D = D=(} a_(11)&a_(12)&a_(13) a_(21)&a_(22)&a_(23) a_(31)&a_(32)&a_(33) ) ,which one is false about D? A. C_(11)=vert } a_(22)&a_(23) a_(32)&a_(33) vert B. dot (M)_(12)=-vert } a_(21)&a_(23) a_(31)&a_(33) vert C. M_(13)=vert } a_(21)&a_(22) a_(31)&a_(32) vert D. c_(32)=-vert } a_(11)&a_(13) a_(21)&a_(23) vert 9. Which one is true about an nxn invertible matrix A? det(A)=det(A^T) B. det(AA^T)=(det(A))^2 C. det(AA^-1)=1 D. det(A^-1)det(A^T)=1 . E 10. If M=(} -3&5 2&-3 is equal to; A. (} 3&5 2&3 ) B. (} -3&-5 -2&-3 ) C、 (} 3&-5 -2&3 ) (} -3&5 2&-3 ) What is the determinant of the matrix N N=(} -1&0&0 -1&-2&0 1&1&8 ) ? A. -32 B. 16 C、 -16 D. 5

### 7. Solution(s) of the given system of linear equations:The system of equations is: Substitute each option into the equations to check if they satisfy both.- **Option A**: \((-1, 1, 1)\): Substituting into : (True). Substituting into : \(-1 - 3(1) + 4(1) = -1 - 3 + 4 = 0\) (True). - **Option B**: \((1, -1, -1)\): Substituting into : (True). Substituting into : \(1 - 3(-1) + 4(-1) = 1 + 3 - 4 = 0\) (True). - **Option C**: \((0.5, -0.5, -0.5)\): Substituting into : (True). Substituting into : \(0.5 - 3(-0.5) + 4(-0.5) = 0.5 + 1.5 - 2 = 0\) (True). Since all options satisfy the equations, the correct answer is: **D. All**---### 8. False statement about matrix :The cofactor and minor definitions are as follows: - \(C_{ij} = (-1)^{i+j} M_{ij}\), where is the determinant of the submatrix obtained by removing the -th row and -th column. Check each option: - **A**: : True. - **B**: : True. - **C**: : True. - **D**: : False. The cofactor should be \((-1)^{3+2} M_{32} = (-1)^5 M_{32} = M_{32}\), not . The correct answer is: **D.**---### 9. True statement about an invertible matrix :- **A**: \(\det(A) = \det(A^T)\): True. Determinants of a matrix and its transpose are equal. - **B**: \(\det(AA^T) = (\det(A))^2\): True. For any square matrix , \(\det(AA^T) = \det(A) \cdot \det(A^T) = \det(A)^2\). - **C**: \(\det(AA^{-1}) = 1\): True. Since , and \(\det(I) = 1\). - **D**: \(\det(A^{-1}) \det(A^T) = 1\): False. \(\det(A^{-1}) = 1/\det(A)\), so \(\det(A^{-1}) \det(A^T) = \frac{\det(A^T)}{\det(A)} = 1\) only if \(\det(A) = 1\), which is not guaranteed for all invertible matrices. The correct answer is: **E. All except D**---### 10. Inverse of matrix :Matrix . The formula for the inverse of a matrix is: where \(\det(M) = ad - bc\). For : Thus: The correct answer is: **C. **---### Determinant of matrix :Matrix . Using cofactor expansion along the first row: The correct answer is: **B. 16**