Which of the following matrices are regular? (a) ( 2 1 1 4 ), (b) ( 0

Which of the following matrices are regular? (a) ( 2 ...

Which of the following matrices are regular? (a) $\left(\begin{array}{ll}2 & 1 \\ 1 & 4\end{array}\right)$, (b) $\left(\begin{array}{ll}0 & -1 \\ 3 & -2\end{array}\right)$, (c) $\left(\begin{array}{rrr}3 & -2 & 1 \\ -1 & 4 & -3 \\ 3 & -2 & 5\end{array}\right)$, (d) $\left(\begin{array}{rrr}1 & -2 & 3 \\ -2 & 4 & -1 \\ 3 & -1 & 2\end{array}\right)$, (e) $\left(\begin{array}{rrrr}1 & 3 & -3 & 0 \\ -1 & 0 & -1 & 2 \\ 3 & 3 & -6 & 1 \\ 2 & 3 & -3 & 5\end{array}\right)$

Which of the following matrices are regular?
(a) $\left(\begin{array}{ll}2 & 1 \\ 1 & 4\end{array}\right)$,
(b) $\left(\begin{array}{ll}0 & -1 \\ 3 & -2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}3 & -2 & 1 \\ -1 & 4 & -3 \\ 3 & -2 & 5\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & -2 & 3 \\ -2 & 4 & -1 \\ 3 & -1 & 2\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}1 & 3 & -3 & 0 \\ -1 & 0 & -1 & 2 \\ 3 & 3 & -6 & 1 \\ 2 & 3 & -3 & 5\end{array}\right)$

Key Concepts

Matrix Inverse

The inverse of a matrix is the matrix that, when multiplied with the original matrix, results in the identity matrix. Finding the inverse is possible if and only if the matrix is regular. The process of computing the inverse often involves calculating the determinant and applying methods like the adjugate matrix method or Gaussian elimination. This concept is pivotal for solving linear equations and understanding linear transformations.

Determinant

The determinant is a scalar value computed from the elements of a square matrix and plays a critical role in determining whether a matrix is regular. If the determinant is nonzero, the matrix is regular (or invertible); if it is zero, the matrix is singular (non-invertible). The determinants provide insight into properties such as volume scaling in transformations represented by the matrix and linear independence of its rows or columns.

Regular Matrix

A regular matrix, also known as an invertible or non?singular matrix, is one that has an inverse. This means that there exists another matrix which, when multiplied by the original matrix, yields the identity matrix. The concept of regularity is central in linear algebra, especially when solving systems of linear equations, as it implies that the system has a unique solution.

Transcript

00:01 In the problem, they have given us three matrix matrices, a, matrix a, matrix c and they have asked us to find which of the matrix is a normal metric.

00:11 Okay, before starting with the problem, let us look what is a normal matrix.

00:17 So let's say if a matrix a is there, okay? how do we know whether it's a normal matrix or not? if a matrix a, if a matrix a, if a into or a into a transpose equal to, a transpose into a.

00:34 If this is the condition, then we call the matrix a as a normal matrix.

00:39 Now, what is the meaning of transpose of a matrix? so transpose of a matrix.

00:42 How to find a transpose of a matrix? to find transpose of a matrix, what we do? we interchange rows and columns.

00:50 For example, let's say a matrix a is there like this.

00:54 A, b, c, d.

00:57 So first row is a and b.

00:59 Second row is c and d.

01:00 So, how to write the transpose of this matrix? transpose is given by a, b here, c, d here.

01:10 So this is called the transpose of the matrix.

01:13 Okay.

01:14 So, this is definition of a normal matrix and this is transpose.

01:19 Okay.

01:20 Now let us move on to our first problem.

01:23 So, three matrices are given.

01:25 So let us deal with matrix a first.

01:27 So matrix a is given as, okay? we'll start with i'll call part a okay so matrix a is given as 3 minus 4 4 okay now to check whether it's a normal matrix matrix or not we'll have to find out the a transpose and multiply okay now first let's write a transpose here a transpose is 3 minus 4 4 and 3.

02:11 So this is a transpose, right? now, let me find a into a transpose first.

02:19 So that will be 3 minus 4, sorry, 4 3...

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