SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 1 - Matrics [Latest edition]

The adjoint matrix of `[(3, -3, 4),(2, -3, 4),(0, -1, 1)]` is 

A = `[(cos alpha, - sin alpha,  0),(sin alpha, cos alpha,  0),(0, 0, 1)]`, then A⁻¹ is

The solution (x, y, z) of the equation `[(1, 0, 1),(-1, 1, 0),(0, -1, 1)] [(x),(y),(z)] = [(1),(1),(2)]` is (x, y, z) =

If ω is a complex cube root of unity, then the matrix A = `[(1, ω^2, ω),(ω^2, ω, 1),(ω, 1, ω^2)]` is

If A = `[(4, -1),(-1, "k")]` such that A² − 6A + 7I = 0, then K = ______

`cos theta [(cos theta, sin theta),(-sin theta, cos  theta)] + sin theta [(sin theta, - cos theta),(cos theta, sin theta)]` = ______

If A = `[(0, 0, -1),(0, -1, 0),(-1, 0, 0)]`, then the only correct statement about the matrix A is ______

If A = `[(cos alpha, sin alpha),(-sin alpha, cos 10 alpha)]`, then A¹⁰ = ______

The element of second row and third column in the inverse of `[(1, 2, 1),(2, 1, 0),(-1, 0, 1)]` is ______

If A = `[(4, 5),(2, 5)]`, then |(2A)⁻¹| = ______

If `[(x - y - z),(-y + z),(z)] = [(0),(5),(3)]`, then the value of x, y and z are respectively ______

The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______

If A = `[(3, 0, 0),(0, 3, 0),(0, 0, 3)]`, then |A| |adj A| = ______

System of equations x + y = 2, 2x + 2y = 3 has ______

If A = `[(1, -1, 1),(2, 1, -3),(1, 1, 1)]`, 10B = `[(4, 2,2),(-5, 0, ∞),(1, -2, 3)]` and B is the inverse of matrix A, then α = ______

If A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find the value of a₃₁A₃₁ + a₃₂A₃₂ + a₃₃A₃₃ 

For an invertible matrix A, if A . (adjA) = `[(10, 0),(0, 10)]`, then find the value of |A|

If the inverse of the matrix `[(alpha, 14, -1),(2, 3, 1),(6, 2, 3)]` does not exists then find the value of α

If A = `[(2, 2),(-3, 2)]` and B = `[(0, -1),(1, 0)]`, then find the matrix (B⁻¹ A⁻¹)⁻¹ 

A = `[(cos theta, - sin theta),(-sin theta, -cos theta)]` then find A⁻¹ 

If A = `[("a", "b"),("c", "d")]` then find the value of |A|⁻¹ 

If A = `[(3, 1),(5, 2)]`, and AB = BA = I, then find the matrix B

If A(α) = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]` then prove that A²(α) = A(2α)

If A = `[(1, 2),(3, -2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, 3)]` then find the order of AB

A + I = `[(3, -2),(4, 1)]` then find the value of (A + I)(A − I)

If A = `[(2, -1, 1),(-2, 3, -2),(-4, 4, -3)]` the find A² 

If A = `[(-2, 4),(-1, 2)]` then find A² 

If A = `[(0, 3, 3),(-3, 0, -4),(-3, 4, 0)]` and B = `[(x),(y),(z)]`, find the matrix B'(AB)

If f(x) = x² − 2x − 3 then find f(A) when A = `[(1, 2),(2, 1)]`

If A = `[(-1),(2),(3)]`, B = `[(3, 1, -2)]`, find B'A'

If A is invertible matrix of order 3 and |A| = 5, then find |adj A|

If A = `[(6, 5),(5, 6)]` and B = `[(11, 0),(0, 11)]` then find A'B'

If A = `[(2, 4),(1, 3)]` and B = `[(1, 1),(0, 1)]` then find (A⁻¹ B⁻¹)

If A = `[(2,0),(0, 1)]` and B = `[(1),(2)]` then find the matrix 𝑋 such that A⁻¹ X = B

Find the matrix X such that AX = I where A = `[(6, 17),(1, 3)]`

Find A^–1 using adjoint method, where A = `[(cos theta, sin theta),(-sin theta, cos theta)]`

Find A⁻¹ using column transformations:

A = `[(5, 3),(3, -2)]`

Find A⁻¹ using column transformations:

A = `[(2, -3),(-1, 2)]`

Find the adjoint of matrix A = `[(6, 5),(3, 4)]`

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

If A = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`, then find A² and hence find A⁻¹ 

If A = `[(0, 1),(2, 3),(1, -1)]` and B = `[(1, 2, 1),(2, 1, 0)]`, then find (AB)⁻¹ 

If A = `[(-4, -3, -3),(1, 0, 1),(4, 4, 3)]`, find adj (A)

Solve the following by inversion method 2x + y = 5, 3x + 5y = −3

If A = `[(1, 2, -1),(3, -2, 5)]`, apply R₁ ↔ R₂ and then C₁ → C₁ + 2C₃ on A

Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices

If A = `[(4,5),(2,1)]`, show that `"A"^-1 = 1/6("A" - 5"I")`.

Find the adjoint of matrix A = `[(2, 0, -1),(3, 1, 2),(-1, 1, 2)]`

Find the matrix X such that `[(1, 2, 3),(2, 3, 2),(1, 2, 2)]` X = `[(2, 2, -5),(-2, -1, 4),(1, 0, -1)]`

Find the inverse of A = `[(sec theta, tan theta, 0),(tan theta, sec theta, 0),(0, 0, 1)]`

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

If A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)]` and B = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find the matrix X such that XA = B

Find the inverse of A = `[(2, -3, 3),(2, 2, 3),(3, -2, 2)]` by using elementary row transformations

If A = `[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`, find A⁻¹ by the adjoint method

Solve the following equations by using inversion method.

x + y + z = −1, x − y + z = 2 and x + y − z = 3

Find the inverse of  A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)]` by elementary column transformations.

If A = `[("x",0,0),(0,"y",0),(0,0,"z")]` is a non-singular matrix, then find A⁻¹ by using elementary row transformations. Hence, find the inverse of `[(2,0,0),(0,1,0),(0,0,-1)]`

Find the inverse of A = `[(costheta, -sintheta, 0),(sintheta, costheta, 0),(0, 0, 1)]` by using elementary row transformations

If A = `[(1,1),(1,2)], "B" = [(4,1),(3,1)]` and C = `[(24,7),(31,9)]`, then find the matrix X such that AXB = C

If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A

If A = `[(2, 3),(1, 2)]`, B = `[(1, 0),(3, 1)]`, find AB and (AB)⁻¹ 

Solve the following system of equations by using inversion method

x + y = 1, y + z = `5/3`, z + x = `4/3`

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices