Please Help me with this Linear Algebra Maths questions

Question

Q.1 Determine the scalars c for which A − c x Identity Matrix [2x2] is not invertible,

where A = [ 4 6

−1 −3 ]

Q7. Let S be the parallelogram determined by the vectors

b1 = [ 2 and b2 = [ 2

3] 5 ]

and let A = [ 6 3 Compute the area of the image of S under the mapping x → Ax.

3 2].

Answer 1

For 1, recall determinant for a 2x2 matrix is ad-bc for the matrix

a b

c d

For 7, compute the determinant of the 2x2 matrix with first column Ab1 and second column Ab2, then take the absolute value. This is because determinant in an n-dimensional vector space corresponds to the n-dimension volume of the parallelipipped spanned by those vectors. In 2 dimensions, this corresponds to the area of the parallelogram between the vectors.