I need help to solve these questions (Question 4 - 7) with working, thank you!

4. For each of the following matrices, (i) find all the eigenvalues of the matrix; (ii) use the answer in (i) to determine whether the matrix is non-singular. (iii) find a basis for the eigenspace associated with each eigenvalue of the matrix; (iv) determine whether the matrix is diagonalizable; (v) find a non-singular matrix P that diagonalize the matrix if it is diagonalizable. 5. Find a 3 x 3 matrix which has eigenvalues 1, 0 and -1 with corresponding eigen- vectors respectively. 6. Given that A is a 3 x 3 matrix such that Without finding A, write down the matrices X and Y such that AX - (198) and 48- ( : !) 7. (MATLAB) The terms yaw, pitch and roll are used in aerospace industry to de- scribe the maneuvering of an aircraft. pitch - y yaw Refering to the diagram above, a yaw is a rotation in the ry-plane. A yaw by an angle e (clockwise) is a linear transformation with standard matrix Y = cos sine o - sine cose 0 0 A pitch is a rotation in the zz-plane. A pitch by an angle o (clockwise) is a linear transformation with standard matrix cosa 0 - sina P = 0 0 sin a 0 COS OF A roll is a rotation in the ye-plane. A roll by an angle S (clockwise) is a linear transformation with standard matrix R = 0 cos B 0 - sin 8 sin B cos B If we perform a yaw by an angle & followed by a pitch by an angle a, the composite transformation is given by the product YP. Similarly, a yaw with angle #, followed by a pitch with angle o, and then a roll with angle # is given by the product Y PR. The effect of such a combination of rotations is again a rotation about a certain axis. This axis is given by an eigenvector of the transformation matrix that corresponds to the eigenvalue 1 (why?). Suppose an aircraft perform a yaw with 20, followed by a pitch with 30, and then a roll with 15. (i) Write down the matrix Q of the composite transformation. (ii) Check that 1 is an eigenvalue of Q. (iii) Find the axis of rotation of the composite transformation. (iv) Is the axis of rotation the same if we perform the roll first, followed by the pitch, and then by the yaw