Matrices — AS & A Level Further Mathematics (9231) questions by topic

The matrix M is given by M = (1 2; 0 1)(cos theta, -sin theta; sin theta, cos theta), where 0 < theta < 2pi. The matrix M represents a sequence of two geometrical transformations…

The matrix M is given by M = ( 1 2 ; 0 1 )( cos(theta) -sin(theta) ; sin(theta) cos(theta) ), where 0 < theta < 2pi. (Here ( 1 2 ; 0 1 ) is the 2x2 matrix with first row (1, 2)…

The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor 14, followed by a rotation anticlockwise about…

It is given that lambda is an eigenvalue of the non-singular square matrix A, with corresponding eigenvector e. Show that e is an eigenvector of A^3 with corresponding eigenvalue…

It is given that lambda is an eigenvalue of the non-singular square matrix A, with corresponding eigenvector e. Show that e is an eigenvector of A^3 with corresponding eigenvalue…

Find the values of a for which the system of equations (3/2)x + 3y + 8z = 1, ax + 3y + 4z = 2, ay - z = 3, does not have a unique solution.

Find the values of k for which the system of equations x + 2y + 3z = 1, kx + 5y + 6z = 2, 7x + 2ky + 9z = 3, does not have a unique solution.

The matrix A is given by A = ( 1 7 11 ; 0 2 5 ; 0 0 -3 ) (a 3x3 matrix with rows [1, 7, 11], [0, 2, 5], [0, 0, -3]). Find a matrix P and a diagonal matrix D such that A^6 = P D…

The matrix M is given by M = ( 1/2, -(1/2)sqrt(3) ; (1/2)sqrt(3), 1/2 )( 14, 0 ; 0, 1 ) (the product of the 2x2 matrix with first row [1/2, -(1/2)sqrt(3)] and second row…

The matrix A is given by A = ( k, 1, 0 ; 6, 5, 2 ; -1, 3, -k ), where k is a real constant. Show that A is non-singular.

The matrix M is given by M = ( 1, 2 ; 0, 1 )( 7, 0 ; 0, 1 ). The matrix M represents a sequence of two geometrical transformations in the x-y plane. Give full details of each…

Find the set of values of a for which the system of equations 6x + ay = 3, 2x - y = 1, x + 5y + 4z = 2 has a unique solution.

The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor k (k != 0), followed by a shear, x-axis fixed,…

The matrices A, B and C are given by A = (1, 2, 3; 2, 1, 3; 3, 2, 5) (the 3x3 matrix with rows [1, 2, 3], [2, 1, 3], [3, 2, 5]), B = (0, -2; -1, 3; 0, 0) (the 3x2 matrix with rows…

The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor k (k != 0), followed by a shear, x-axis fixed,…

Find the set of values of k for which the system of equations x + 5y + 6z = 1, kx + 2y + 2z = 2, -3x + 4y + 8z = 3, has a unique solution and interpret this situation…

The matrix A is given by A = ( -11, 1, 8; 0, -2, 0; -16, 1, 13 ). Show that ( 1, 1, 1 )^T is an eigenvector of A and state the corresponding eigenvalue.

The matrix A is given by A = ( -2 0 0 ; 0 7 9 ; 4 1 7 ). Show that the characteristic equation of A is lambda^3 - 12 lambda^2 + 12 lambda + 80 = 0 and find the eigenvalues of A.

Find the set of values of k for which the system of equations x + 5y + 6z = 1, kx + 2y + 2z = 2, -3x + 4y + 8z = 3, has a unique solution and interpret this situation…

The matrix A is given by A = [ -11 1 8 ; 0 -2 0 ; -16 1 13 ]. Show that the column vector (1, 1, 1)^T is an eigenvector of A and state the corresponding eigenvalue.

Let A be the 2x2 matrix A = [[3, 0], [1, 1]] (first row (3, 0), second row (1, 1)). Prove by mathematical induction that, for all positive integers n, 2A^n = [[2 3^n, 0], [3^n -…

The matrix M is given by M = [[a, b^2], [c^2, a]] (first row (a, b^2), second row (c^2, a)), where a, b, c are real constants and b != 0. Show that M does not represent a rotation…

Let A = ( 3, 0 ; 1, 1 ) (the 2x2 matrix with first row [3, 0] and second row [1, 1]). Prove by mathematical induction that, for all positive integers n, 2A^n = ( 23^n, 0 ; 3^n -…

The matrix M is given by M = ( a, b^2 ; c^2, a ) (the 2x2 matrix with first row [a, b^2] and second row [c^2, a]), where a, b, c are real constants and b != 0. Show that M does…

The matrix M is given by M = ( cos 2theta, -sin 2theta ; sin 2theta, cos 2theta )( 1, k ; 0, 1 ), where 0 < theta < pi and k is a non-zero constant. The matrix M represents a…

Show that the system of equations x + 2y + 3z = 1, 4x + 5y + 6z = 1, 7x + 8y + 9z = 1, does not have a unique solution.

The matrix A is given by A = ( 18 5 -11 ) ( 8 6 -4 ) ( 32 10 -20 ) Show that the characteristic equation of A is lambda^3 - 4 lambda^2 - 20 lambda + 48 = 0 and hence find the…

Show that the system of equations x + 2y + 3z = 1, 4x + 5y + 6z = 1, 7x + 8y + 9z = 1, does not have a unique solution.

The matrix A is given by A = ( 18 5 -11 ; 8 6 -4 ; 32 10 -20 ) (that is, the rows of A are (18, 5, -11), (8, 6, -4) and (32, 10, -20)). Show that the characteristic equation of A…

The matrix A is given by A = [ [a, -6a, 2a+2], [0, 1-a, 0], [0, 2-a, -1] ], where a is a constant with a != 0 and a != 1. Show that the equation A (x, y, z)^T = (1, 2, 3)^T has a…

Let k be a constant. The matrices A, B and C are given by A = [[1, k, 3], [2, 1, 3], [3, 2, 5]] (rows (1, k, 3), (2, 1, 3), (3, 2, 5)), B = [[0, -2], [-1, 3], [0, 0]] (rows (0,…

The matrix M is given by M = ( k, 0 ; 0, 1 )( 1, 0 ; 1, 1 ), where k is a constant and k != 0 and k != 1. (Here M is the product of the 2x2 matrix with first row [k, 0], second…

Let k be a constant. The matrices A, B and C are given by A = ( 1, k, 3 ; 2, 1, 3 ; 3, 2, 5 ), B = ( 0, -2 ; -1, 3 ; 0, 0 ) and C = ( -2, -1, 1 ; 1, 1, 3 ). It is given that A is…

Show that the system of equations 14x - 4y + 6z = 5, x + y + kz = 3, -21x + 6y - 9z = 14, where k is a constant, does not have a unique solution and interpret this situation…

The matrix A is the 3x3 upper-triangular matrix with rows (-6, 2, 13), (0, -2, 5) and (0, 0, 8), i.e. A = ( -6 2 13 ; 0 -2 5 ; 0 0 8 ). Find a matrix P and a diagonal matrix D…

The matrix P is given by P = ( 1 -1 1 ) ( 0 2 1 ) ( 0 0 -1 ). State the eigenvalues of P.

Show that the system of equations 14x - 4y + 6z = 5, x + y + kz = 3, -21x + 6y - 9z = 14, where k is a constant, does not have a unique solution and interpret this situation…

The matrix A is the 3x3 upper-triangular matrix with rows (-6, 2, 13), (0, -2, 5) and (0, 0, 8). Find a matrix P and a diagonal matrix D such that A^(-1) = P D P^(-1).