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Matrix Determinant

Have you ever considered how to find out if a system of simultaneous equations contains a solution? You may set up a system of equations to compare deals on something you're looking to buy to compare several factors across the different options, but how do you check if you've set up the system correctly and if there are solutions to compare?

You can store this system in a matrix and then find the determinant of this matrix to dictate if you have a solution available.

What Is A Matrix?

A matrix is an array used to store, display and compute data. The internals are called elements and the matrix will have $$m$$ columns and $$n$$ rows.

To understand what a determinant is and how to apply it, we must first understand what a matrix is.

A matrix is a way of displaying information- for example, a system of simultaneous equations can be written in matrix form where your columns are for a variable and rows are for your equations. The solutions would then form a column vector. The matrix notation makes it easier to carry out transformations and solve data sets- especially when there are more than 2 equations to solve!

But how do we go about solving a matrix? That's where determinants come in - we use them to solve matrices.

The general matrix notation is that $$m$$ denotes the number of columns and $$n$$ denotes the number of rows. The internals of the matrix can then be written as:$A_{m,n} = \begin{bmatrix}a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\\vdots & \vdots & \ddots & \vdots \\a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix}$

$$2\times 2$$ matrix example$A_{2,2}=\begin{bmatrix}3&7\\12&-3\end{bmatrix}$

$$2\times 3$$ matrix example$A_{2,3}=\begin{bmatrix}2&-4&19\\11&23&5\end{bmatrix}$

$$4\times 3$$ matrix example$A_{4,3}=\begin{bmatrix}2&8&4\\-2&-5&-3\\13&9&7\\-7&3&-2\end{bmatrix}$

Matrices are a really useful way of displaying and storing a lot of information and they are widely used throughout mathematics, physics and engineering at the higher levels of those disciplines.

What is A Determinant?

So now we know the general basics of matrices but what is a determinant and why is it relevant?

The determinant is a value we can figure out for any square matrix that we can then use to calculate the inverse matrix.

A square matrix is a matrix that has an equal number of rows and columns, $$m=n.$$

As you can see below, square matrices have an equal number of rows and columns to form a square shape

$$2\times 2$$ matrix example $A_{2,2}=\begin{bmatrix}3&7\\12&-3\end{bmatrix}$

$$3\times 3$$ matrix example $A_{3,3}=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}$

An invertible matrix is a matrix for which we can find another matrix such that their product is the identity matrix $$(I)$$.

Our initial matrix can be denoted as $$A$$ and the second matrix is the inverse of this matrix so is denoted as $$A^{-1}$$. This gives the identity $AA^{-1}=I.$ You can think of inverse matrices as the reciprocals of the matrix world.

The determinant also tells us if a matrix is invertible. Let the determinant of matrix A be denoted as $$\det{A}.$$

• If the determinant of the matrix A is given as $$\det{A} = 0$$ the matrix is singular and therefore does not have an inverse. There is no solution to this matrix.
• If the determinant of the matrix A is given as $$\det{A} \neq 0$$ the matrix is non-singular and therefore does have an inverse. There is a solution to this matrix.

For more information and examples on inverting a matrix see our article Inverting Matrices.

Determinant Of 2x2 Matrix

So now we know what a determinant is and what it's used for but we still need to find out how they work.

Let's start with the most basic form - a $$2\times 2$$ matrix determinant. The method to calculate the determinant of a $$2\times 2$$ matrix is most basically explained by cross multiplying and then subtraction of these multiplied values.

Let's consider the below matrix,$A_{2,2} = \begin{bmatrix}a_{1,1} & a_{1,2} \\a_{2,1} & a_{2,2}\end{bmatrix}$This is the notation we used before but lets write it with distinct elements so the methodology is easier to follow. Therefore,$A_{2,2} = \begin{bmatrix}a&b\\c&d\end{bmatrix}$Our first step is to cross multiply- we multiply top left, bottom right and then top right, bottom left- and then subtract the second multiplication from the first. Therefore,$\det{A}=ad-cb$In our original notation this would be,$\det{A}=a_{1,1}a_{2,2}-a_{1,2}a_{2,1}$Let's now apply this to an example.

Find the determinant of matrix $$A$$ below and then identify if the matrix is invertible.$A=\begin{bmatrix}4&9\\-2&8\end{bmatrix}$

Solution

Step 1. Find the determinant

\begin{align} \det{A}&=ad-cb\\&=(4\cdot 8)-(9\cdot -2)\\&=32-(-18)\\&=50\end{align}

Step 2. Identify if matrix $$A$$ is invertible

$$\det{A} \neq 0$$ so the matrix $$A$$ is non-singular and as such invertible.

we move next to learn how to find the determinant of a $$3\times 3$$ matrix.

Determinant of 3x3 Matrix

Now we've seen how to find the determinant of a $$2\times 2$$ matrix but we can also encounter $$3\times 3$$ matrices in Further Maths so let's look at how to find the determinants of these now.

The process is slightly more complex than a $$2\times 2$$ matrix determinant but follows the same principles. Let's consider the matrix below,$A_{3,3}=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$The way we calculate the determinant of this is to break it down into a series of $$2\times 2$$ matrices.

To do this we go along the top row and multiply each of the elements there by the determinant of its minor.

The minor in a $$3\times 3$$ matrix are the elements left if you cross out the row and column stemming from your root element.

Let's say our root element to start with is $$a$$. The minor would be the $$2\times 2$$ matrix left, if we cross out all the values horizontally and vertically from this element.$\begin{bmatrix}a&-&-\\|&e&f\\|&h&i\end{bmatrix}$The determinant of our $$3\times 3$$ matrix considers all of the minors for the top row and results in the below formula.\begin{align}\det{A}&=a\cdot \det{\begin{bmatrix}e&f\\h&i\end{bmatrix}}-b\cdot \det{\begin{bmatrix}d&f\\g&i\end{bmatrix}}+c\cdot \det{\begin{bmatrix}d&e\\g&h\end{bmatrix}} \\ & =a(ei-fh)-b(di-fg)+c(dh-eg)\end{align}

Note the sign convention here with the determinant formula - it goes $$+,-,+$$.

In the tip above you can see the sign convention goes $$+,-,+$$. These are the cofactors for the first row of a $$3\times 3$$ matrix.

While slightly beyond the scope you are expected to deal with here, there are cofactors for every element in the matrix.

This means we could also find the determinant of a matrix using rows 2 or 3 as our root elements and take the minors from there- we would just need to apply the correct cofactors to do so.

However, all you need to concern yourself with for now is the top row and $$+,-,+$$.

Let's now have a look at how to apply this to an example.

Find the determinant of the below matrix.$A_{3,3}=\begin{bmatrix}4&8&12\\7&19&2\\0&5&2\end{bmatrix}$

Solution

We apply our formula for the determinant.\begin{align}\det{A}&=a(ei-fh)-b(di-fg)+c(dh-eg)\\&=4[(19\cdot 2)-(2\cdot 5)]-8[(7\cdot 2)-(2\cdot 0)]+12[(7\cdot 5)-(19\cdot 0)]\\&=4[(38)-(10)]-8[(14)-(0)]+12[(35)-(0)]\\&=4(28)-8(14)+12(35)\\&=112-112+420\\&=420 \end{align}

We move now to know more the determinant of a diagonal matrix, after defining it.

Determinant Of A Diagonal Matrix

To calculate the determinate of a diagonal matrix, we must first understand what one is.

A diagonal matrix is a matrix that has all non-diagonal elements as 0. This doesn't mean that the diagonal elements themselves can't hold the value of 0 but it does mean any non-diagonal element is 0.

It takes the form of,$A=\begin{bmatrix}a_{1,1} & 0 &0& \cdots & 0 \\0 & a_{2,2} & 0&\cdots & 0 \\0&0&a_{3,3}&\cdots &0\\\vdots & \vdots &\vdots & \ddots & \vdots \\0 & 0 &0& \cdots & a_{m,n} \end{bmatrix}$

The determinant of a diagonal matrix can be found while multiplying the diagonal elements.

The determinant of a diagonal matrix is the product of the diagonal elements. Therefore,$\det{A}=a_{1,1}\cdot a_{2,2}\cdot a_{3,3} \cdot \quad \cdots \quad \cdot a_{m,n}$

If the diagonal elements are not all non-zero values then the matrix cannot be non-singular as a product with a $$0$$ in will always return a solution of $$0$$ and as we've seen previously this makes the matrix singular and uninvertible.

Let's have a look at this in an example.

Find $$\det{A}$$, where,$A_{5,5}=\begin{bmatrix}13&0&0&0&0\\0&-6&0&0&0\\0&0&7&0&0\\0&0&0&-1&0\\0&0&0&0&3\end{bmatrix}$

Solution

We know the determinant of a diagonal matrix is the product of the diagonal elements. \begin{align} \det{A}&=a_{1,1}\cdot a_{2,2}\cdot a_{3,3}\cdot a_{4,4}\cdot a_{5,5}\\&=(13)\cdot (-6)\cdot (7)\cdot (-1)\cdot (3)\\&=1638.\end{align}

Can we calculate the determinant of an inverse matrix? The answer is, YES!

Determinant Of An Inverse Matrix

Our final matrix to consider when looking at determinants is that of the inverse matrix.

For the inverse matrix to exist, we know the original matrix must have had a determinant that was a non-zero value. We also earlier likened the inverse matrix to a reciprocal of the original matrix- that is going to come into play again here.

The determinant of an inverse matrix is equal to the inverse or reciprocal of the original matrix. In mathematical terms this means that the determinant of an inverse matrix takes the following form,$\det{A^{-1}}=\frac{1}{\det{A}}.$

Let's take the follwoing example.

Take matrix $$A$$ below and find if it is invertible. If matrix $$A$$ is invertible, find the determinant of that inverse matrix.

$A=\begin{bmatrix}6&2\\12&9\end{bmatrix}$

Solution

Step 1. Find the determinant of $$A$$

\begin{align} \det{A}&=ad-cb\\&=(6\cdot 9)-(12\cdot 2)\\&=54-24\\&=30\end{align}

Step 2. Identify if matrix $$A$$ is invertible

$$\det{A} \neq 0$$ so the matrix $$A$$ is non-singular and as such invertible.

Step 3. Find the determinant of the inverse matrix

\begin{align}\det{A^{-1}}&=\frac{1}{\det{A}}\\&=\frac{1}{30}. \end{align}

Matrix Determinant - Key takeaways

• The determinant of a $$2\times 2$$ matrix is given by,$\det{A}=ad-bc$
• The determinant of a $$3\times 3$$ matrix uses the top row elements and the determinate of their respective minors to calculate the determinant. This is given by,\begin{align}\det{A}&=a\cdot \det{\begin{bmatrix}e&f\\h&i\end{bmatrix}}-b\cdot \det{\begin{bmatrix}d&f\\g&i\end{bmatrix}}+c\cdot \det{\begin{bmatrix}d&e\\g&h\end{bmatrix}} \\ & =a(ei-fh)-b(di-fg)+c(dh-eg)\end{align}
• The determinant of a diagonal matrix is the product of all the diagonal elements- which are the only elements that can be non-zero. This is given by, $\det{A}=a_{1,1}\cdot a_{2,2}\cdot a_{3,3} \cdot \quad \cdots \quad \cdot a_{m,n}$
• The determinant of an inverse matrix is the inverse of the determinant of the original matrix. This is given by, $\det{A^{-1}}=\frac{1}{\det{A}}$

Times the top row elements of the matrix by the 2x2 determinant of its minor. You will then follow the sign convention of +,-,+ for putting these three minor and 2x2 determinants together and this will be your 3x3 determinant.

Multiply the top left element by the bottom right element. Then take from this the top right element multiplied by the bottom left element.

If the matrix is 2x2 then multiply the top left element by the bottom right element. Then take from this the top right element multiplied by the bottom left element. If 3x3 then take the top row of elements, times them by the determinant of their minors and follow the sign convention of +,-,+ for putting these together.

Yes- they will both have the same determinant

The determinant of the matrix will be a non-zero value

Final Matrix Determinant Quiz

Question

What is a matrix?

A 2D array used to store, display and compute data.

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Question

How is the size of a matrix defined?

By m rows and n columns.

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Question

To find the determinant of a matrix what size\shape must it be?

Square

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Question

If a matrix is singular what does this mean?

The determinant is 0 and therefore the matrix is not invertible.

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Question

If a matrix is non-singular what does this mean?

The determinant is a non-zero value and the matrix is therefore invertible.

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Question

How do you find the determinant of a 2x2 matrix?

Multiply the top left element by the bottom right element. Then take from this the top right element multiplied by the bottom left element.

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Question

What is a minor in a 3x3 matrix?

The elements left if you cross out the row and column stemming from your root element.

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Question

How do you find the determinant of a 3x3 matrix?

Take the top row of elements, times them by the determinant of their minors and follow the sign convention of +,-,+ for putting these together.

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Question

What is the determinant of an inverse matrix?

The reciprocal of the determinant of the original matrix.

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Question

What is a diagonal matrix?

A matrix where any non-diagonal elements are 0

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Question

What is the determinant of a diagonal matrix?

The product of the diagonal elements

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Question

Can diagonal elements in a diagonal matrix hold the value of 0?

Yes

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Question

What does this mean for the determinant if a diagonal matrix has a 0 value in the diagonal elements?

Since the determinant is the product of the diagonal elements the determinant will be 0 so therefore, the matrix will be non-invertible.

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Question

How can we write the determinant of a 3x3 matrix in formula form?

det(A)=a(ei-fh)-b(di-fg)+c(dh-eg)

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Question

How can we write the determinant of a 2x2 matrix in formula form?

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