Test: Linear Algebra

• If λ₁ ,λ₂ ,λ₃ ....λ_n are the eigen values of a non–singular matrix A, then A^-1 has the eigen values  1/λ₁ ,1/λ₂ ,1/λ₃ ....1/λ_n 
• Thus eigen values of A^-1are 1/2, 1/3, -1/3

• If λ₁ ,λ₂ ,λ₃ ....λ_n are the eigen values of  a matrix A, then A² has the eigen values  λ₁² ,λ₂² ,λ₃² ....λ_n² 
• So, eigen values of A² are 1, 4, 9.

• Since the sum of the eigenvalues of an n–square matrix is equal to the trace of the matrix (i.e. sum of the diagonal elements)
• So, required sum = 8 + 5 + 5  = 18

Since, the eigenvalues of A and A^T are square so the eigenvalues of A^T are 2 and 4.

• Since 1 and 3 are the eigenvalues of A so the characteristic equation of A is:
• Also, by Cayley–Hamilton theorem, every square matrix satisfies its own characteristic equation. So:
  

Since the product of the eigenvalues is equal to the determinant of the matrix so: |A| = 1 x 2 x 5 = 10

• Since A + B is defined, A and B are matrices of the same type, say m x n. Also, AB is defined.
• So, the number of columns in A must be equal to the number of rows in B i.e. n = m.
• Hence, A and B are square matrices of the same order.

Inverse matrix is defined for square matrix only.

A must be invertible.

For a set of linear equations, Ax = b. The inverse of matrix A exists (i.e. |A| ≠ 0).

This is the necessary condition for the existence of a solution for this system.

In each row (as well as each column), the third figure is a combination of all the elements of the first and the second figures.

Determinant of a skew-symmetric even ordered matrix A should be a perfect square.

For orthogonal matrix

From linear algebra for A_nxn triangular matrix . DetA = The product of the diagonal entries of A.