Solved Examples on Similar Matrices

Example 1: Find the matrix B if A and B are similar matrices. If

.

Solution:

We know that if A and B are similar matrices, then P^-1AP = B.

P^-1 = Adj P/|P|

|P| = 0 × 3 − (1 × 2) = -2

Example 2: Prove that the determinants of two similar matrices are the same.

Solution:

Let us consider two similar matrices A and B, to prove that their determinants are the same.

We know that if A and B are similar matrices, then P^-1AP = B, where P is the change-of-basis matrix.

Now, det (B) = det (P^-1AP)

⇒ det (B) = det (P^-1) det (A) det (P)

We know that det (P^-1) = 1/det (P)

So, det (B) = 1/det (P) × det (A) × det (P)

⇒ det (B) = 1 × det (A)

⇒ det (B) = det (A)

Hence, the determinants of two similar matrices A and B are the same.

Example 3: Prove that the matrix given below is similar to itself.

Solution:

To prove that a matrix is similar to itself, we have to prove that A = I^-1AI,

where the identity matrix “I” is the change-of-basis matrix.

I^-1 = Adj I/ det (I)

Hence proved.

Example 4: Verify whether the matrices given below are similar or not.

.

Solution:

To prove that two matrices are similar, we have to prove that B = P^-1AP.

P^-1 = Adj P/|P|

As B ≠ P^-1AP, A and B are not similar.

A “matrix” is referred to as a rectangular array of numbers that are arranged in rows and columns. The horizontal lines are said to be rows, while the vertical lines are said to be columns. We can determine the size of a matrix by the number of rows and columns in it. If a matrix has “m” rows and “n” columns, then it is said to be an “m by n” matrix and is written as an “m × n” matrix.

For example, a matrix with five rows and three columns is a “5 × 3” matrix. We have various types of matrices, like rectangular, square, triangular, symmetric, singular, etc. In this article, we learn about similar matrices, their examples, and their properties.