Determinants quiz

The value of the determinant of [[2, 4], [-1, 2]] is:

If A is a 3 x 3 matrix with det A = 5, then det(2A) is:

For any square matrix A, det(A transpose) equals:

A square matrix A is invertible if and only if:

The cofactor of the element 2 in [[1, 2], [3, 4]] is:

If A is a 3 x 3 matrix, det(adj A) equals:

If det A = 4 for a 3 x 3 matrix A, then det(A^-1) is:

If three points are collinear, the area of the triangle formed by them is:

If det [[x, 2], [18, x]] = det [[6, 2], [18, 6]], then x is:

The system AX = B has a unique solution when:

Evaluate the determinant of [[cos theta, -sin theta], [sin theta, cos theta]]. (2 marks)

Find the minors and cofactors of all elements of [[2, -4], [0, 3]]. (2 marks)

If A = [[3, 1], [2, 2]], find adj A. (2 marks)

Find the value of x for which the matrix [[x, 4], [2, 2]] is singular. (2 marks)

Show that the points (1, 2), (3, 6) and (5, 10) are collinear using determinants. (2 marks)

If A is an invertible matrix of order 2 with det A = 3, find det(adj A). (2 marks)

Write the formula for the area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) in determinant form. (2 marks)

If A and B are invertible matrices of the same order, prove that (AB)^-1 = B^-1 A^-1. (2 marks)

Is the system x + 2y = 2 and 2x + 3y = 3 consistent? Justify. (2 marks)

If A is a square matrix with A^2 = A, find det A in the possible cases (assume det A exists). (2 marks)

Evaluate the determinant of [[3, -1, -2], [0, 0, -1], [3, -5, 0]]. (3 marks)

Find the inverse of A = [[1, 2], [3, 4]]. (3 marks)

Using determinants, check whether the points (1, -1), (2, 1) and (4, 5) are collinear. (3 marks)

Using determinants, find the equation of the line joining (1, 2) and (3, 6). (3 marks)

Solve 5x + 2y = 4 and 7x + 3y = 5 by the matrix method. (3 marks)

Read the passage and answer the questions. A school awards prizes for discipline, sincerity and punctuality. The total amount for one prize of each is Rs 6,000. Three times the punctuality prize added to the sincerity prize is Rs 11,000, and the discipline prize plus the punctuality prize is twice the sincerity prize. (Amounts in thousands: x, y, z.) (i) Write the system of equations in matrix form. (ii) Find det A. (iii) Solve to find the value of each prize. (5 marks)

Read the passage and answer the questions. A farmer owns a triangular plot with corners at (0, 0), (8, 0) and (4, 6), measured in units of 10 m. (i) Write the determinant for its area. (ii) Find the area in square units. (iii) Find the area in square metres. (5 marks)

Read the passage and answer the questions. A stationery shop sells pens (Rs x), notebooks (Rs y) and files (Rs z). Customer P buys one of each for Rs 60. Customer Q buys 2 pens, 1 notebook and 3 files for Rs 130. Customer R buys 1 pen, 2 notebooks and 1 file for Rs 80. (i) Write the equations in matrix form AX = B. (ii) Show that A is non-singular. (iii) Find x, y and z. (5 marks)

Read the passage and answer the questions. A data encryption system multiplies a message matrix by A = [[2, 3], [1, 2]]. To decode, the receiver needs A^-1. (i) Find det A. (ii) Find A^-1. (iii) Decode the encoded column [8, 5] by computing A^-1 times it. (5 marks)

Read the passage and answer the questions. A student claims that the system x + y + z = 3, 2x + 2y + 2z = 6 and 3x + 3y + 3z = 10 has a unique solution because there are three equations in three unknowns. (i) Find the determinant of the coefficient matrix. (ii) Is the student correct? (iii) Is the system consistent? Explain. (5 marks)

Solve x + y + z = 6, x + 2y - 3z = -4 and 2x - y + 3z = 9 using the matrix method. (6 marks)

Solve 2x - 3y + 5z = 11, 3x + 2y - 4z = -5 and x + y - 2z = -3 using the matrix method. (6 marks)

Find A^-1 for A = [[1, -1, 1], [2, 1, -3], [1, 1, 1]] and hence solve x - y + z = 4, 2x + y - 3z = 0 and x + y + z = 2. (6 marks)

If A = [[2, 3], [1, -4]] and B = [[1, -2], [-1, 3]], verify that (AB)^-1 = B^-1 A^-1. (6 marks)

Find the area of the triangle with vertices (2, 7), (1, 1) and (10, 8) using determinants, and draw the triangle. Find the equation of the side joining (1, 1) and (10, 8) using determinants. (6 marks)