Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Verify A(adj A) = (adj A)A = |A|I.

`[(2,3),(-4,-6)]`

Verify A(adj A) = (adj A)A = |A|I.

`[(1,-1,2),(3,0,-2),(1,0,3)]`

Find the inverse of the matrices (if it exists).

`[(2,-2),(4,3)]`

Find the inverse of the matrices (if it exists).

`[(-1,5),(-3,2)]`

Find the inverse of the matrices (if it exists).

`[(1,2,3),(0,2,4),(0,0,5)]`

Find the inverse of the matrices (if it exists).

`[(1,0,0),(3,3,0),(5,2,-1)]`

Find the inverse of the matrices (if it exists).

`[(2,1,3),(4,-1,0),(-7,2,1)]`

Find the inverse of the matrices (if it exists).

`[(1,-1,2),(0,2,-3),(3,-2,4)]`

Find the inverse of the matrices (if it exists).

`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`

Let A = `[(3,7),(2,5)]` and B = `[(6,8),(7,9)]`. Verify that (AB)⁻¹ = B⁻¹A⁻¹.

If A = `[(3,1),(-1,2)]` show that A² – 5A + 7I = 0. Hence, find A^–1.

For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A² + aA + bI = 0.

For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A³ − 6A² + 5A + 11 I = 0. Hence, find A⁻¹.

If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A³ − 6A² + 9A − 4I = 0 and hence find A⁻¹.

If A is an invertible matrix of order 2, then det (A⁻¹) is equal to ______.

If A⁻¹ = `[(3,-1,1),(-15,6,-5),(5,-2,2)]` and B = `[(1,2,-2),(-1,3,0),(0,-2,1)]`, find (AB)⁻¹.

Let A = `[(1,2,1),(2,3,1),(1,1,5)]` verify that

1. [adj A]^–1 = adj(A^–1)
2. (A^–1)^–1 = A

If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.

Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ ≤ 2π, then ______.

Show that the three lines with direction cosines `12/13, (-3)/13, (-4)/13;  4/13, 12/13, 3/13;  3/13, (-4)/13, 12/13 ` are mutually perpendicular.