Problem 252 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find all idempotent diagonal matrices of order 3 .

Step-by-Step Solution

Answer

The eight 3x3 idempotent, diagonal matrices are all possible combinations of 0 and 1 on the diagonal.

Step 1: Defining the idempotency condition

Start by noting that a 3x3 diagonal matrix has the form \( A = \begin{{bmatrix}} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{{bmatrix}} \). We can immediately write down the square of this matrix, as it's also a diagonal matrix, with the entries being the square of the individual diagonal elements of the original matrix, i.e., \( A^2 = \begin{{bmatrix}} a^2 & 0 & 0 \\ 0 & b^2 & 0 \\ 0 & 0 & c^2 \end{{bmatrix}} \). Since A is idempotent, we have the condition that \( A^2 = A \).

Step 2: Solving the idempotency condition

The idempotency condition \( A^2 = A \) is then a system of equations for the elements of the matrix: \( a^2 = a, b^2 = b, c^2 = c \). Solving these gives the possible values for the diagonal elements.

Step 3: Find possible values for the diagonal elements

The equations \( x^2 = x \) each have two solutions: \( x = 0 \) and \( x = 1 \). Therefore, for each of a, b, and c, there are two choices, giving a total of \( 2^3 = 8 \) different matrices.

Key Concepts

idempotent matrices

Idempotent matrices are quite intriguing in the world of linear algebra. An idempotent matrix is a matrix that remains unchanged when squared. In more formal terms, if you have a matrix \( A \), it is idempotent if:

• \( A^2 = A \)

This means that when you multiply the matrix by itself, you end up with the same matrix you started with.
For diagonal matrices, which have nonzero elements only on the diagonal while the other elements are zero, the idempotency condition simplifies quite a bit. Each diagonal element \( x \) must satisfy the equation \( x^2 = x \). Therefore, each diagonal element can only be 0 or 1.
When we consider a diagonal matrix of order 3, we're essentially looking at choices for each of its three diagonal elements. Since each element has two possible values (0 or 1), you end up with a total of 8 different idempotent matrices of this order. It's fascinating how such a simple condition leads to such clear and concise solutions.

matrix order

The order of a matrix is a straightforward but important concept to grasp. It refers to the number of rows and columns in a matrix. For example, a \(3 \times 3\) matrix has 3 rows and 3 columns.
In our given exercise, we focused on a \(3 \times 3\) matrix, which is a square matrix. Square matrices are particularly interesting because they have the same number of rows and columns. Such matrices often have special properties, like being able to be idempotent. Diagonal matrices also arise naturally as square matrices where all the non-diagonal entries are zero.
Calculating properties and performing operations on matrices depend heavily on their order. Higher order matrices involve more complex calculations but follow similar basic rules applicable to all matrices.

system of equations

Solving systems of equations is a core technique in algebra that is very useful when analyzing matrices. When a condition like idempotency is imposed on a matrix, it often results in a system of equations involving its elements.
For a \(3 \times 3\) idempotent matrix, as seen in our exercise, we derive the following system of equations by enforcing \( A^2 = A \):

• \( a^2 = a \)
• \( b^2 = b \)
• \( c^2 = c \)

Each of these is a simple quadratic equation that can have two solutions: \( x = 0 \) or \( x = 1\). As each diagonal entry must satisfy these equations, solving them gives us the possible values of the matrix's diagonal entries. Thus, understanding and solving systems of equations allows us to uncover the characteristics and possible configurations of matrices.