A square matrix P is called idempotent if P^2=P. (a) Find all 2 ×2 idempotent upper triangular m

step 1 Welcome to this lesson. In this lesson, we'll show that a, which is a matrix, a 3 by 3 matrix is

A square matrix P is called idempotent if P^2=P. (a) Find...

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Key Concepts

Idempotent Matrix

An idempotent matrix is a square matrix that, when multiplied by itself, yields the same matrix. This property is formalized by the equation P^2 = P. Idempotent matrices are closely related to projection operators in linear algebra, where the matrix projects vectors onto a subspace and repeated applications have no further effect.

Upper Triangular Matrix

An upper triangular matrix is a square matrix in which all the entries below the main diagonal are zero. These matrices simplify many linear algebra problems because operations such as multiplication and finding determinants become easier due to the structure of zeros in the lower part of the matrix.

Matrix Equation Solving

Solving matrix equations involves setting up equations by equating corresponding entries of matrices that are stated to be equal. For a problem like finding all matrices satisfying a condition like P^2 = P, one expands the left-hand side through matrix multiplication and then equates each entry to the corresponding entry of P, resulting in a system of algebraic equations that must be solved simultaneously.

Transcript

00:01 Welcome to this lesson.

00:03 In this lesson, we'll show that a, which is a matrix, a 3 by 3 matrix is id important.

00:13 All right.

00:14 So let's multiply a by itself.

00:17 And if it's equals to a, then we have shown it.

00:24 So this, you have negative 1, 0, negative 1.

00:28 Now this squared actually means multiplying it by itself.

00:33 So let me open a very big curly brackets.

00:39 So this is how we do matrix multiplication.

00:42 It goes this way and that way.

00:44 So we have two times two, which is four.

00:48 Then plus we have zero times one, which is zero.

00:54 Then we have two times negative one, which is negative two.

00:59 All right then again would have two times zero which is zero then zero times one which is zero again then two times zero which is zero all right then again we have two times two which is four then zero times negative one it's negative all right then let's come here we have 1 times 2 which is 2 then 1 times 1 which is 1 and we have 2 times negative 1 which is negative okay then 1 times 0 which is 0 which is 0 which is 1 which is positive 1 and 2 times 0 then we have 1 times 2, which is 2.

02:03 1 times 2 again, which is 2.

02:06 Then we have 2 times negative 1, which is negative 2.

02:11 Then let's come to the last rule...

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