A diagonal matrix is a square matrix whose elements, various other than the diagonal, are zero. There are specific problems that should be met for a matrix to be referred to as a diagonal matrix. Firstly, let’s check the formal definition of a diagonal matrix.

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A square matrix in which all the facets other than the major diagonal are zero is recognized as a diagonal matrix.

In this write-up, we are going to take a cshed look at what provides a matrix diagonal, exactly how to find diagonal matrices, properties of diagonal matrices, and also the determinant of a diagonal matrix. Let’s start!

What is a Diagonal Matrix?

A matrix to be classified as a diagonal matrix, it has to fulfill the complying with conditions:

square matrixall facets (entries) of the matrix, other than the principal diagonal, hregarding be $ 0 $

A square matrix is said to be:

lower triangular if its aspects over the major diagonal are all $ 0 $top triangular if its facets listed below the major diagonal are all $ 0 $

Lower Triangular Matrix

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diagonal matrix is a unique square matrix that is BOTH upper and lower triangular considering that all aspects, whether above or listed below the primary diagonal, are $ 0 $.

How to uncover Diagonal Matrix

To uncover, or recognize, a diagonal matrix, we must watch if it is a square matrix and all the elements besides the primary diagonal (diagonal that runs from optimal left to bottom right) are $ 0 $. Let’s take a look at the matrices presented below:

$ A = eginpmatrix 3 & 0 \ 0 & – 3 finish pmatrix $

$ B = eginpmatrix 1 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & – 2 end pmatrix $

$ C = eginbmatrix 10 & 0 \ 0 & 12 \ 0 & 12 end bmatrix $

$ D = eginbmatrix – 5 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 9 finish bmatrix $

Note the following observations about each of the $ 4 $ matrices shown above:

Matrix $ A $ is a square matrix bereason its has actually the exact same variety of rows and also columns ($ 2 imes 2 $ matrix). The principal diagonal has entries $ 3 $ and $ -3 $, respectively. All other entries are $ 0 $. Thus, this is a diagonal matrix.Matrix $ B $ is a square matrix because its has the very same number of rows and columns ($ 3 imes 3 $ matrix). The primary diagonal has entries $ 1 $, $ 5 $, and $ -2 $, respectively. All various other entries are $ 0 $. Hence, this is a diagonal matrix.A false glance at Matrix $ C $ will certainly make us think that it is a diagonal matrix. But, first and also foremany, it is not a square matrix. Hence, it cannot be a diagonal matrix.This one’s a little bit tricky. First of all, it is in truth a square matrix ( $ 3 imes 3 $ ). You will certainly think that it is not a diagonal matrix.But why? Is it bereason there’s a $ 0 $ in the middle?

If you take a closer look at the interpretation of a diagonal matrix, you will view that nowright here does it say that the entries in the diagonal cannot be $ 0 $! The condition is that the aspects various other than the diagonal hregarding be $ 0 $. So, also if tbelow are elements that are $ 0 $ in the diagonal, it won’t issue. As lengthy as the elements besides the diagonal are $ 0 $, it will certainly be a diagonal matrix.

Hence, Matrix $ D $ is in reality a diagonal matrix!

This brings us to $ 2 $ special types of diagonal matrices:

Identity MatrixZero Matrix

Identity Matrix

This is a square matrix in which all the entries in the major diagonal are $ 1 $ and also all other facets are $ 0 $. A $ 2 imes 2 $ and a $ 3 imes 3 $ identification matrices are shown below.

$ 2 imes 2 $ identity matrix

$ eginbmatrix 1 & 0 \ 0 & 1 finish bmatrix $

$ 3 imes 3 $ identification matrix

$ eginbmatrix 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end bmatrix $

You can read even more about identity matrices below.

Zero Matrix

A matrix in which all the elements are $ 0 $. A $ 2 imes 2 $ and a $ 3 imes 3 $ zero matrices are shown below.

$ 2 imes 2 $ zero matrix

$ eginbmatrix 0 & 0 \ 0 & 0 end bmatrix $

$ 3 imes 3 $ zero matrix

$ eginbmatrix 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 finish bmatrix $

Now, let’s look at some properties of diagonal matrices.

Properties of Diagonal Matrices

Tright here are numerous properties of diagonal matrices but for the objective of this article, we will look at $ 3 $ properties of diagonal matrices. Below, we take a look at the properties and their examples.

Property 1:

When $ 2 $ diagonal matrices of the very same order are included or multiplied together, the resultant matrix is another diagonal matrix through the exact same order.

Consider the matrices shown below:

$ A = eginbmatrix 3 & 0 \ 0 & 4 finish bmatrix $$ B = eginbmatrix 1 & 0 \ 0 & 5 finish bmatrix $

Now, we add both the $ 2 imes 2$ matrices and also present that the resultant matrix is likewise diagonal.

$ A + B = eginbmatrix 3+1 & 0+0 \ 0+0 & 4+5 finish bmatrix $

$ A + B = eginbmatrix 4 & 0 \ 0 & 9 finish bmatrix $

Hence, we check out that the resultant matrix, $A + B$, is additionally a diagonal matrix of the order $2 imes 2$.

Let’s check matrix multiplication via the same matrices. We multiply Matrix $A$ and also Matrix $B$ and show that the resultant is additionally a diagonal matrix via the very same order. Shown below:

$ A imes B = eginbmatrix 3 & 0 \ 0 & 4 finish bmatrix imes eginbmatrix 1 & 0 \ 0 & 5 end bmatrix $

$ A imes B = eginbmatrix 3 imes1+0 imes0 & 3 imes 0 + 0 imes 5 \ 0 imes 1 + 4 imes 0 & 0 imes0 + 4 imes 5 finish bmatrix $

$ A imes B = eginbmatrix 3 & 0 \ 0 & 20 endbmatrix $

Therefore, we see that the resultant matrix, $A imes B$, is likewise a diagonal matrix of the order $2 imes 2$.

To learn even more about how we did matrix multiplication, please have actually a look at the write-up below.

Property 2:

The transpose of a diagonal matrix is the matrix itself.

If we have a matrix $A$, then we signify its transpose as $A^T$. Transposing a matrix implies to flip its rows and columns. Let’s present that this residential property is true by calculating the transpose of Matrix $A$.

$ A = eginbmatrix 3 & 0 \ 0 & 4 end bmatrix $

$ A^T = eginbmatrix 3 & 0 \ 0 & 4 end bmatrix $

Intertransforming the rows and columns produces the very same matrix because of the entries besides the diagonal being $0$.

Property 3:

Under multiplication, diagonal matrices are commutative. 

If we have $2$ matrices, $A$ and also $B$, this implies $AB = BA$. Let’s show this building by using the two matrices from over.

$ A imes B = eginbmatrix 3 imes1+0 imes0 & 3 imes 0 + 0 imes 5 \ 0 imes 1 + 4 imes 0 & 0 imes0 + 4 imes 5 end bmatrix $

$ A imes B = eginbmatrix 3 & 0 \ 0 & 20 endbmatrix $

Now,

$ B imes A = eginbmatrix 1 imes3+0 imes0 & 1 imes 0 + 0 imes 4 \ 0 imes 3 + 5 imes 0 & 0 imes0 + 5 imes 4 end bmatrix $

$ B imes A = eginbmatrix 3 & 0 \ 0 & 20 endbmatrix $

Therefore, we have actually seen that $AB = BA$.

Determinant of Diagonal Matrix

First, let’s look at the determinant of a $2 imes2$ matrix.

Consider Matrix $M$ presented below:

$ M = eginpmatrix a & b \ c & d endpmatrix $

The determinant of this matrix is:

$ det(M) = ad – bc $

One home of a diagonal matrix is that the determinant of a diagonal matrix is equal to the product of the aspects in its major diagonal.

Let’s see if it’s true by finding the determinant of the diagonal matrix displayed listed below.

$ N = eginpmatrix 2 & 0 \ 0 & 8 endpmatrix $

$ det(N) = (2 imes8)-(0 imes0) = 16 $

This is in reality the product of the elements in its diagonal, $2 imes8=16$.

Example 1

For the matrices displayed listed below, recognize whether they are diagonal matrix or not.

$ A = eginbmatrix -2 & 0 \ 0 & -7 end bmatrix $

$ B = eginbmatrix a & 0 \ 0 & b \ 0 & d end bmatrix $

$ C = eginbmatrix 3 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 11 finish bmatrix $

$ D = eginbmatrix 0 & 0 \ 0 & 0 end bmatrix $

Solution

Matrix A is a $2 imes2$ matrix via the elements being 0 other than the diagonal. So, this is a diagonal matrix.Matrix B is a $3 imes2$ matrix. It’s not square, so automatically we can say that it is not a diagonal matrix.Matrix C is a square matrix ($3 imes3$). Also all the aspects besides the diagonal are $0$. So, it is a diagonal matrix. Furthermore, an enattempt of the diagonal is likewise $0$, it doesn’t matter as lengthy as all the entries except the diagonal are zeros.Matrix D is a one-of-a-kind kind of diagonal matrix. It is a zero matrix. Thus, it is a diagonal matrix.Example 2

Will multiplying Matrix A and also Matrix B cause a diagonal matrix?

$ A = eginbmatrix -9 & 0 \ 0 & 0 finish bmatrix $

$ B = eginbmatrix 1 & 0 \ 1 & 1 finish bmatrix $

Solution

Matrix A is a diagonal matrix however Matrix B is not. So, multiplying Matrix A and B will certainly not result in a diagonal matrix.

Example 3

Find the determinant of the matrix displayed below:

$ B = eginbmatrix 8 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & -1 end bmatrix $

Solution

Matrix B is a $3 imes3$ diagonal matrix. Respeak to that the product of all the entries of the diagonal of a diagonal matrix is its determinant. Therefore, we sindicate multiply and discover the answer:

$ det(B) = 8 imes 4 imes -1 = – 32$

Practice Questions

Identify which of the complying with matrices are diagonal matrices.$ J = eginpmatrix 0 & 0 \ 0 & -5 endpmatrix $$ K = eginpmatrix 0 & 2 \ 1 & -1 endpmatrix $$ L = eginbmatrix -3 & 0 & 0 \ 0 & -5 & 0 \ 0 & 0 & 3 endbmatrix $Calculate the determinant of the matrix displayed below:$ T = eginbmatrix -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 4 endbmatrix $Given$ A = eginpmatrix 2 & 0 \ 0 & -1 endpmatrix $$ B = eginpmatrix 1 & 0 \ 1 & -2 endpmatrix $

Is $AB = BA$ ?

Answers

Matrix J is a square matrix. All the elements other than the major diagonal are zeros. This is a diagonal matrix.Matrix K is a square matrix yet not all the aspects, except the diagonal, are zero. Thus, this is not a diagonal matrixMatrix L is a square matrix (3 imes3). The facets various other than the diagonal entries are zero. So, this is a diagonal matrix.This is a diagonal matrix. We deserve to uncover the determinant of this matrix by taking the product of all the 3 entries of the diagonal. Thus, the determinant is:$ det(T) = -1 imes -1 imes 4 = 4 $If two matrices are diagonal, the multiplication of those two matrices are commutative. Looking at Matrix A and also B, we deserve to check out that Matrix A is diagonal however Matrix B is not. Hence, their multiplication will certainly not be commutative.

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Hence, $ AB eq BA$.