1. State the problem: Write 20 standard multiple choice questions on determinant, minor, cofactor, inverse, equality of matrix, symmetry and skew symmetric, operation on matrix. 2. Since the user requests multiple choice questions rather than a single math problem, I will provide 20 questions covering the requested topics. 3. Each question will test understanding of key concepts such as determinant calculation, minor and cofactor definitions, inverse matrix conditions, matrix equality, symmetric and skew symmetric properties, and matrix operations. 4. This approach ensures comprehensive coverage of the topics for learning and assessment. 5. Here are the 20 multiple choice questions: 1. What is the determinant of a 2x2 matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$? A) $ad + bc$ B) $ad - bc$ C) $ab - cd$ D) $ac - bd$ 2. The minor of an element in a matrix is: A) The determinant of the matrix B) The determinant of the submatrix formed by deleting the element's row and column C) The cofactor of the element D) The transpose of the matrix 3. The cofactor of an element is: A) Its minor multiplied by $(-1)^{i+j}$ where $i,j$ are the element's row and column B) The determinant of the matrix C) The element itself D) The inverse of the element 4. A matrix is invertible if and only if: A) Its determinant is zero B) Its determinant is non-zero C) It is symmetric D) It is skew symmetric 5. The inverse of a matrix $A$ satisfies: A) $AA^{-1} = I$ B) $A + A^{-1} = I$ C) $A - A^{-1} = 0$ D) $A^{-1} = A$ 6. Two matrices $A$ and $B$ are equal if: A) They have the same determinant B) They have the same size and corresponding elements are equal C) They are both symmetric D) They have the same trace 7. A symmetric matrix satisfies: A) $A = -A^T$ B) $A = A^T$ C) $A^2 = I$ D) $A$ is diagonal 8. A skew symmetric matrix satisfies: A) $A = A^T$ B) $A = -A^T$ C) $A$ is invertible D) $A$ is diagonal 9. The determinant of a skew symmetric matrix of odd order is: A) Always zero B) Always one C) Non-zero D) Undefined 10. The transpose of a product of matrices $AB$ is: A) $A^T B^T$ B) $B^T A^T$ C) $AB$ D) $BA$ 11. The determinant of a product of matrices $AB$ is: A) $det(A) + det(B)$ B) $det(A) det(B)$ C) $det(A) - det(B)$ D) $det(A) / det(B)$ 12. The minor of element $a_{ij}$ in matrix $A$ is calculated by: A) Deleting row $i$ and column $j$ and finding determinant of remaining matrix B) Multiplying $a_{ij}$ by $(-1)^{i+j}$ C) Transposing $A$ D) Adding all elements except $a_{ij}$ 13. The cofactor matrix is: A) Matrix of minors B) Matrix of cofactors C) Transpose of matrix D) Inverse of matrix 14. The adjoint of a matrix is: A) The transpose of the cofactor matrix B) The inverse matrix C) The determinant D) The matrix itself 15. The inverse of a matrix $A$ can be found by: A) $\frac{1}{det(A)}$ times adjoint of $A$ B) Transpose of $A$ C) Matrix of minors D) Matrix of cofactors 16. If $A$ is symmetric, then $A + A^T$ is: A) Symmetric B) Skew symmetric C) Zero matrix D) Identity matrix 17. If $A$ is skew symmetric, then $A + A^T$ is: A) Symmetric B) Skew symmetric C) Zero matrix D) Identity matrix 18. The sum of two symmetric matrices is: A) Symmetric B) Skew symmetric C) Zero matrix D) Identity matrix 19. The product of two symmetric matrices is: A) Always symmetric B) Symmetric if they commute C) Skew symmetric D) Zero matrix 20. The determinant of the transpose of a matrix is: A) Equal to determinant of the matrix B) Negative of determinant C) Square of determinant D) Zero Final answer: 20 multiple choice questions covering determinant, minor, cofactor, inverse, equality, symmetry, skew symmetry, and matrix operations.