Determinants

Determine the values of a for which the matrix M is singular.

Given that matrix M is non-singular and that

• det(M) = −108

• a 0

determine the matrix M⁻¹

Three planes are defined by the following equations

where p and q are constants.

Given that the planes form a sheaf, determine

the value of p

the value of q

Show that M is non-singular for all values of a.

Determine, in terms of a, M^–1

Show that M is non-singular for all values of a.

Determine, in terms of a, M^–1

Prove by mathematical induction that, for n

Triangle T has vertices A, B and C.

Triangle T is transformed to triangle T′ by the transformation represented by Mⁿ where n

Given that

• triangle T has an area of 5 cm²

• triangle T′ has an area of 1215 cm²

• vertex A (2, –2) is transformed to vertex A ′(123, –2)

determine

the value of n

the value of a

Prove by mathematical induction that, for n

Triangle T has vertices A, B and C.

Triangle T is transformed to triangle T′ by the transformation represented by Mⁿ where n

Given that

• triangle T has an area of 5 cm²

• triangle T′ has an area of 1215 cm²

• vertex A (2, –2) is transformed to vertex A ′(123, –2)

determine

the value of n

the value of a

The transformation P is an enlargement, centre the origin, with scale factor k, where k 0

The transformation Q is a rotation through angle θ degrees anticlockwise about the origin.

The transformation P followed by the transformation Q is represented by the matrix

Determine

the value of k,

the smallest value of θ

A square S has vertices at the points with coordinates (0, 0), (a, –a), (2a, 0) and (a, a) where a is a constant.

The square S is transformed to the square S′ by the transformation represented by M.

Determine, in terms of a, the area of S′

The transformation P is an enlargement, centre the origin, with scale factor k, where k 0

The transformation Q is a rotation through angle θ degrees anticlockwise about the origin.

The transformation P followed by the transformation Q is represented by the matrix

Determine

the value of k,

the smallest value of θ

A square S has vertices at the points with coordinates (0, 0), (a, –a), (2a, 0) and (a, a) where a is a constant.

The square S is transformed to the square S′ by the transformation represented by M.

Determine, in terms of a, the area of S′

A is a 2 by 2 matrix and B is a 2 by 3 matrix.

Giving a reason for your answer, explain whether it is possible to evaluate

AB

A + B

Given that

where a, b and λ are constants,

determine

• the value of λ

• the value of a

• the value of b

Given that

1	1	1
0	sin	cos

determine the values of θ for which the matrix M is singular.

A is a 2 by 2 matrix and B is a 2 by 3 matrix.

Giving a reason for your answer, explain whether it is possible to evaluate

AB

A + B

Given that

where a, b and λ are constants,

determine

• the value of λ

• the value of a

• the value of b

Given that

determine the values of θ for which the matrix M is singular.

Given that k ≠4, find, in terms of k, the inverse of the matrix M.

Find, in terms of p, the coordinates of the point where the following planes intersect.

2 x + 5 y + 7 z = 1 x + y + z = p 2 x + y − z = 2

Find the value of q for which the following planes intersect in a straight line.

4 x + 5 y + 7 z = 1 x + y + z = q 2 x + y − z = 2

For this value of q, determine a vector equation for the line of intersection.

Given that k ≠4, find, in terms of k, the inverse of the matrix M.

Find, in terms of p, the coordinates of the point where the following planes intersect.

2 x + 5 y + 7 z = 1 x + y + z = p 2 x + y − z = 2

Find the value of q for which the following planes intersect in a straight line.

4 x + 5 y + 7 z = 1 x + y + z = q 2 x + y − z = 2

For this value of q, determine a vector equation for the line of intersection.

where k is a constant.

Show that, in terms of k, a characteristic equation for M is given by

Given that det M = 5

find the value of k

use the Cayley‑Hamilton theorem to find the inverse of M.