__ 9. After applying R_(2)arrow R_(2)-2R_(1) to A=(} 1&2 2&-1 ) we get A (} 1&2 2&-5 ) (} 1&2 0&-5 ) CI (} 1&4 2&-3 ) D (} 2&-1 1&2 ) __ 10. What is the solution set of system of linear equations, ) 2x+7y=-5 5x+11y=2 -(5)/(3),(25)/(21) B (69)/(13),-(29)/(13) (5)/(3),-(25)/(21) D -(69)/(13),(29)/(13) __ A=(} 1&2 2&3 ? 11. Suppose A/ (} 3&6 6&9 ) If X is a matrix such that (} 3&3 3&3 ) D then (} 3&1 1&3 ) __ 12. The Augmented matrix associated to the system of equations ) 2x+y+3z=4 x-z=1 -4x+y-3=0 A (} 3&2&1&1 -1&1&-1&3 -1&-4&1&4 ) __ 13. The inverse of the matrix (} 2&3 3&4 ) __ 3) is A (} -4&3 3&-2 ) B/( (} 4&-3 -3&2 ) Cl (} -2&3 3&-4 ) D/ (} -4&-3 -3&-2 ) __ 14. The dimension of the augmented matrix is 4times 6 The corresponding system of linear equations has how many variables and how many equations? A/ 4 variables, 6 equations B/6 variables, 4 equations C/5 variables, 4 equations D/6 variables, 3 equations __ 15. If A is a square matrix of order m, then the matrix B of the same order is called the inverse of the matrix A, if (Assume that I is an identity matrix of order m) A AB=A BA=A Cl AB=BA=1 AB=BA __ 16. If A is a 3times 3 matrix and det (A)=5 , then how much is det (2A^TA) A/100 B /200 C/50 D/20 __ 17. The solution set of the system of linear equation ) x+y+2z=4 2x+3y+3z=5 3x+3y+7z=14 is __ A (-1,1,2) B/ (1,2,3) CI (1,-1,2) D (2,1,-2) A=(} -3&2&5 1&1&4 2&1&0 ) is __ __ 18. The determinant of the matrix A A B/23 C/25 D -25 -23 __ 19. Let A = A=(} 1&-1&3 -2&0&2 4&5&-4 ) then the cofactor of 2 is __ (A) 1 (B) -1 (C) 9 (D) -9 __ 20. Which one of the following is true about the matrix. (} 1&3&5 6&4&2 9&7&0 ) ? A/ The minor of the entry 3 is 18. B/ The cofactor of the entry 5 is -6 C/ The minor of the entry 0 is 14. D/ The cofactor of the entry 1 is -14 2

# Explanation:## Step 1: Apply the row operation to matrix A### We need to apply the row operation to the given matrix . This means we subtract 2 times the first row from the second row. Thus, the new matrix is .# Answer:### B. # Explanation:## Step 1: Write the augmented matrix for the system of equations### The system of linear equations is given by: We write the augmented matrix as: ## Step 2: Use Gaussian elimination to solve the system### Perform row operations to convert the matrix into row-echelon form:1. Multiply the first row by : 2. Subtract 5 times the first row from the second row: 3. Multiply the second row by : 4. Subtract times the second row from the first row: ## Step 3: State the solution### The solution is the last column of the new matrix: Simplifying these fractions, we get: # Answer:### D. # Explanation:## Step 1: Set up the equation involving matrices### Given and , we need to find matrix .## Step 2: Calculate ### The transpose of is: ## Step 3: Substitute and simplify the equation### Substitute and into the equation: This simplifies to: Since is invertible, multiply both sides by : where is the identity matrix: # Answer:### B/C. # Explanation:## Step 1: Write the augmented matrix for the system of equations### The system of linear equations is given by: We write the augmented matrix as: # Answer:### D. # Explanation:## Step 1: Find the inverse of the given matrix### The matrix is given by: The formula for the inverse of a matrix is: For our matrix: Calculate the determinant: Thus, the inverse is: # Answer:### A. # Explanation:## Step 1: Determine the number of variables and equations### The dimension of the augmented matrix is . This means there are 4 rows (equations) and 6 columns. Since one column is for the constants, the remaining 5 columns represent the variables.# Answer:### C. 5 variables, 4 equations# Explanation:## Step 1: Define the condition for the inverse matrix### For a matrix to be the inverse of matrix , the product of and must equal the identity matrix of the same order: # Answer:### D. # Explanation:## Step 1: Calculate the determinant of the given matrix expression### Given that is a matrix with , we need to find .Using properties of determinants: Since has the same determinant as : Thus: # Answer:### B. 200# Explanation:## Step 1: Write the augmented matrix for the system of equations### The system of linear equations is given by: We write the augmented matrix as: ## Step 2: Use Gaussian elimination to solve the system### Perform row operations to convert the matrix into row-echelon form:1. Subtract 2 times the first row from the second row: 2. Subtract 3 times the first row from the third row: 3. Add the second row to the third row: ## Step 3: State the solution### The system is inconsistent because the last row represents the equation , which is a contradiction. Therefore, there is no solution.# Answer:### None of the options are correct; the system has no solution.# Explanation:## Step 1: Calculate the determinant of the given matrix### The matrix is given by: The determinant of a matrix is calculated as: Substitute the values: # Answer:### None of the options are correct; the determinant is 23.# Explanation:## Step 1: Calculate the cofactor of the given element### The matrix is given by: We need to find the cofactor of the element 2 in the second row and third column. The cofactor is given by: where is the minor matrix obtained by deleting the -th row and -th column. For the element 2 at position : Calculate the determinant of : Thus, the cofactor is: # Answer:### D. -9# Explanation:## Step 1: Verify the statements about the matrix### The matrix is given by: We need to verify each statement:1. The minor of the entry 3 is the determinant of the submatrix obtained by deleting the first row and second column: So, the minor of the entry 3 is -18, not 18.2. The cofactor of the entry 5 is: So, the cofactor of the entry 5 is 6, not -6.3. The minor of the entry 0 is the determinant of the submatrix obtained by deleting the third row and third column: So, the minor of the entry 0 is -14, not 14.4. The cofactor of the entry 1 is: So, the cofactor of the entry 1 is -14.# Answer:### D. The cofactor of the entry 1 is -14