Explainer: Three-by-Three Determinants

In this explainer, we will learn how to evaluate 3Γ—3 determinants using the cofactors (Laplace expansion) or the Sarrus method.

When working with a square matrix, we will often be interested in calculating the determinant in order to gain potentially useful information about the matrix. As well as being able to tell whether or not the matrix is invertible, the determinant provides information about the physical transformation of space that is encoded by the matrix in question. There are many other reasons that could justify calculating the determinant of a matrix but part of the appeal is the wide range of mathematical properties that are satisfied by the determinant and the variety of ways in which this quantity can be calculated.

This explainer will be split into two parts, with the first section providing a quick reminder of how to calculate the determinant of a 2Γ—2 matrix. Then, in the second part, we will explain how to calculate the determinant of a square matrix which has an order of 𝑛×𝑛, by incorporating the method known for 2Γ—2 matrices. The process that we will show is a generalization of Sarrus’ rule for the determinant of 3Γ—3 matrices and is extendable to any square matrix.

Definition: Determinant of a 2 Γ— 2 Matrix

For a 2Γ—2 matrix 𝐴=ο”π‘Žπ‘π‘π‘‘ο , the β€œdeterminant” of 𝐴 is denoted |𝐴| and is given by the formula |𝐴|=|||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.

When calculating the determinant of a 2Γ—2 matrix, the most common errors typically occur if one of the entries is negative. In particular, if either 𝑏 or 𝑐 is negative, then the formula |𝐴|=π‘Žπ‘‘βˆ’π‘π‘ introduces a risk of a sign error in the rightmost term. In the following examples, we will show how the determinant is calculated for 3 different matrices, with the final example involving a matrix that has entries which are trigonometric functions, instead of numbers.

Example 1: Determinant of a 2 Γ— 2 Matrix

Find the determinant of the following matrix: 5105.

Answer

We label the matrix as 𝐴=5105.

Then, the determinant is calculated as |𝐴|=||5105||=5Γ—5βˆ’1Γ—0=25.

We have therefore found that |𝐴|=25.

Example 2: Determinant of a 2 Γ— 2 Matrix

Find the determinant of the following matrix: 51βˆ’15.

Answer

We denote 𝐴=51βˆ’15.

Then, we calculate |𝐴|=||51βˆ’15||=5Γ—5βˆ’1Γ—(βˆ’1)=25+1.

The answer is then |𝐴|=26.

Example 3: 2 Γ— 2 Determinants with Trigonometric Functions

Find the value of |||||βˆ’1πœƒ11+πœƒβˆ’1πœƒ|||||.sincotsin

Answer

We define the matrix 𝐴=βŽ‘βŽ’βŽ’βŽ£βˆ’1πœƒ11+πœƒβˆ’1πœƒβŽ€βŽ₯βŽ₯⎦.sincotsin

We calculate the determinant as normal: |𝐴|=|||||βˆ’1πœƒ11+πœƒβˆ’1πœƒ|||||=ο€Όβˆ’1πœƒοˆΓ—ο€Όβˆ’1πœƒοˆβˆ’1Γ—ο€Ή1+πœƒο…=1πœƒβˆ’ο€Ή1+πœƒο….sincotsinsinsincotsincot

Although this is technically a correct form for |𝐴|, it is likely that an equivalent expression exists which has a simpler form. Because of this, we will attempt to simplify the expression above. By recalling that cotcossinπœƒ=πœƒπœƒ, we can write that |𝐴|=1πœƒβˆ’ο€Ύ1+πœƒπœƒοŠ.sincossin

To achieve a common denominator across all terms, we can also write 1=πœƒπœƒsinsin to find |𝐴|=1πœƒβˆ’ο€Ύ1+πœƒπœƒοŠ=1πœƒβˆ’ο€Ώπœƒπœƒ+πœƒπœƒο‹=1πœƒβˆ’ο€Ώπœƒ+πœƒπœƒο‹=1βˆ’ο€Ίπœƒ+πœƒο†πœƒ.sincossinsinsinsincossinsinsincossinsincossin

Finally, we know that sincosοŠ¨οŠ¨πœƒ+πœƒ=1, which gives |𝐴|=1βˆ’ο€Ίπœƒ+πœƒο†πœƒ=1βˆ’1πœƒsincossinsin which means that |𝐴|=0. This expression is equivalent to the form of the determinant that we originally calculated but is much more concise. The determinant being equal to zero in particular means that the original matrix is not invertible, irrespective of the value of πœƒ.

With the determinant of 2Γ—2 matrices being well understood, it is now possible to calculate the determinant for larger square matrices. Although specific rules do exist for the determinants of 3Γ—3 and 4Γ—4 matrices, it is better to develop a more general approach, so that this can be applied in a way which minimizes the number of calculations that we have to complete.

Definition: Matrix Minors

Consider a matrix 𝐴 with order π‘šΓ—π‘›. Then, the matrix β€œminor” 𝐴 is the initial matrix 𝐴 after having removed the 𝑖th row and the 𝑗th column. This means that 𝐴 is a matrix with order (π‘šβˆ’1)Γ—(π‘›βˆ’1).

It is easiest to illustrate this concept by example. Suppose that we have a 3Γ—3 matrix 𝐴=ο˜βˆ’23βˆ’3βˆ’4360βˆ’39.

We will begin by calculating the matrix minor 𝐴, which will mean removing the second row and the third column from 𝐴, as highlighted: 𝐴=ο˜βˆ’23βˆ’3βˆ’4360βˆ’39.

Then, all uncolored entries remain in the matrix minor: 𝐴=ο”βˆ’230βˆ’3.

If we wish to calculate the matrix minor 𝐴, then we would be removing the third row of 𝐴 as well as the first column. The highlighted entries in 𝐴=ο˜βˆ’23βˆ’3βˆ’4360βˆ’39 will need to be removed. The resulting matrix minor is 𝐴=3βˆ’336.

This definition can equally be applied to 4Γ—4 matrices or any square matrix of a higher dimension, as we will demonstrate later in this explainer.

Definition: General Formula for the Determinant

Consider a square matrix 𝐴 with order 𝑛×𝑛. Then, the determinant of 𝐴 is calculated in one of two ways, both of which involve the determinant of particular matrix minors of 𝐴. We can choose to calculate the determinant by using a particular row 𝑖 and matrix minors as follows: |𝐴|=ο„š(βˆ’1)π‘Ž|𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+β‹―+(βˆ’1)π‘Ž|𝐴|.οŠο…οŠ²οŠ§οƒοŠ°ο…οƒο…οƒο…οƒοŠ°οŠ§οƒοŠ§οƒοŠ§οƒοŠ°οŠ¨οƒοŠ¨οƒοŠ¨οƒοŠ°οŠοƒοŠοƒοŠ

Alternatively, we can choose one particular column 𝑗: |𝐴|=ο„š(βˆ’1)π‘Ž|𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+β‹―+(βˆ’1)π‘Ž|𝐴|.οŠοƒοŠ²οŠ§οƒοŠ°ο…οƒο…οƒο…οŠ§οŠ°ο…οŠ§ο…οŠ§ο…οŠ¨οŠ°ο…οŠ¨ο…οŠ¨ο…οŠοŠ°ο…οŠο…οŠο…

This definition seems a little daunting at first, although it is much easier to apply than might initially seem apparent. Much of the confusion can be eradicated by approaching the calculations in a consistent and clear manner, highlighting the relevant entries and results as an inherent part of the method.

Example 4: 3 Γ— 3 Determinants

Find the determinant of the matrix 123322098.

Answer

We have a 3Γ—3 matrix that we will label as 𝐴=123322098.

We can choose to calculate |𝐴| by first selecting either one row or one column. For reasons that will be explained shortly, a good strategy is normally to find the row or column which contains the most number of zeros. For our matrix 𝐴, this would be either the third row or the first column, which both contain one zero entry in the position π‘ŽοŠ©οŠ§. We choose the first column, as highlighted: 𝐴=123322098.

These entries are π‘Ž=1, π‘Ž=3, and π‘Ž=0. Now we calculate the corresponding matrix minors: 𝐴=2298,𝐴=2398,𝐴=2322.

Using the method for calculating the determinant of a 2Γ—2 matrix that we used earlier in this explainer, we find that |𝐴|=βˆ’2,|𝐴|=βˆ’11,|𝐴|=βˆ’2.

The final preparation step is to calculate the values (βˆ’1)=1,(βˆ’1)=βˆ’1,(βˆ’1)=1.

We know that we can expand along column 1 using the formula |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|.

It is now clear that we did not need to calculate |𝐴| because this value will be multiplied by π‘Ž=0. Writing out the calculation in full gives |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|=(1)Γ—(1)Γ—(βˆ’2)+(βˆ’1)Γ—(3)Γ—(βˆ’11)+(1)Γ—(0)Γ—(βˆ’2)=βˆ’2+33+0.

This gives the final result |𝐴|=31.

We will complete another example for a 3Γ—3 matrix, although this time with an advantage being gained by expanding along a row, rather than a column as in the previous example. The technique is very similar to that used above and, using the same formatting of our working, there is little difference between this example and the previous one in terms of layout and the calculations involved.

Example 5: 3 Γ— 3 Determinants

Calculate |𝐴| when 𝐴=30βˆ’1010224.

Answer

In order to simplify the subsequent calculations, we identify the row or column that contains the most number of entries which are zero. In this case, this is the second row of 𝐴, as we have highlighted: 𝐴=30βˆ’1010224.

The relevant entries are π‘Ž=0, π‘Ž=1, and π‘Ž=0. Two of these entries are zero, which means that we will not need to calculate two matrix minors, as their determinants will eventually be multiplied by zero. For completeness, we give all relevant matrix minors: 𝐴=0βˆ’124,𝐴=3βˆ’124,𝐴=3022.

The determinants of these matrix minors are |𝐴|=2,|𝐴|=14,|𝐴|=6.

The final ingredient is to calculate the values (βˆ’1)=βˆ’1,(βˆ’1)=1,(βˆ’1)=βˆ’1.

We can calculate the determinant of 𝐴 by expanding along row 2 using the formula |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|.

We see that there are two matrix minors that we did not need to consider, since their determinants are being multiplied by a zero term. Nonetheless, we calculate |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|=(βˆ’1)Γ—(0)Γ—(2)+(1)Γ—(1)Γ—(14)+(βˆ’1)Γ—(0)Γ—(6).

This gives the answer |𝐴|=14.

The techniques that we used in the question above are easy to extend to matrices of a higher order. For a matrix 𝐴 with order π‘šΓ—π‘›, the matrix minor 𝐴 will have order (π‘šβˆ’1)Γ—(π‘›βˆ’1). For example, by repeating the steps from above, the determinant of a 4Γ—4 matrix can be expressed in terms of particular matrix minors that are all of order 3Γ—3.

Example 6: Determinant of 4 Γ— 4 Matrix

Find the determinant of the matrix ⎑⎒⎒⎣1232132241501212⎀βŽ₯βŽ₯⎦.

Answer

We set the above matrix equal to 𝐴. Then, either the third row or the fourth column of 𝐴 would be good choices to expand along, since they both contain one entry which has a value of zero. With no reason to discriminate between these two choices, we arbitrarily select the third row: 𝐴=⎑⎒⎒⎣1232132241501212⎀βŽ₯βŽ₯⎦.

The highlighted entries are π‘Ž=4, π‘Ž=1, π‘Ž=5, and π‘Ž=0οŠͺ. The matrix minors corresponding to these entries are 𝐴=232322212,𝐴=132122112,𝐴=122132122,𝐴=123132121.οŠͺ

We are now in a position to calculate the determinants of these matrix minors. Given that π‘Ž=0οŠͺ, in reality there is no need to calculate |𝐴|οŠͺ, although we will do so regardless for the sake of completion. The results are |𝐴|=βˆ’4,|𝐴|=0,|𝐴|=0,|𝐴|=βˆ’2.οŠͺ

It is fortunate that two of the matrix minors have determinants which are zero, as this will simplify the resultant calculation for |𝐴|. The final preparation step provides the values (βˆ’1)=1,(βˆ’1)=βˆ’1,(βˆ’1)=1,(βˆ’1)=βˆ’1.οŠͺ

We will calculate |𝐴| by expanding along row 3: |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|.οŠͺοŠͺοŠͺ

Using all of the highlighted values above, we have |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|=(1)Γ—(4)Γ—(βˆ’4)+(βˆ’1)Γ—(1)Γ—(0)+(1)Γ—(5)Γ—(0)+(βˆ’1)Γ—(0)Γ—(βˆ’2).οŠͺοŠͺοŠͺ

The presence of so many zeros means that we can quickly deduce that |𝐴|=βˆ’16.

We will provide one more example of a 4Γ—4 matrix, for the sake of demonstrating the technique. At this stage, it might seem as though the technique is still a little mysterious, which is common to many concepts in linear algebra when they are initially studied. However, with practice the technique becomes fairly simple to apply. Furthermore, once a mathematician is familiar with all of the tricks for calculating determinants, there are normally shortcuts that can be taken to get to the answer. The following example provides another demonstration as to how the calculation can be streamlined by a correct choice of row or column.

Example 7: 4 Γ— 4 Determinants

Calculate |𝐴| when 𝐴=⎑⎒⎒⎣14βˆ’50022βˆ’3βˆ’11412βˆ’300⎀βŽ₯βŽ₯⎦.

Answer

The fourth column contains the most number of zeros from any row or column, so we will expand along this column by first highlighting the relevant entries: 𝐴=⎑⎒⎒⎣14βˆ’50022βˆ’3βˆ’11412βˆ’300⎀βŽ₯βŽ₯⎦.

The highlighted entries are π‘Ž=0οŠͺ, π‘Ž=βˆ’3οŠͺ, π‘Ž=1οŠͺ, and π‘Ž=0οŠͺοŠͺ. Since π‘Ž=π‘Ž=0οŠͺοŠͺοŠͺ, there is no need to calculate the matrix minors 𝐴οŠͺ or 𝐴οŠͺοŠͺ, nor their determinants. The appropriate matrix minors are 𝐴=𝐴,𝐴=14βˆ’5βˆ’1142βˆ’30,𝐴=14βˆ’50222βˆ’30,𝐴=𝐴.οŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺ

From this, we use the techniques for calculating the determinant of a 3Γ—3 matrix to find |𝐴|=|𝐴|,|𝐴|=39,|𝐴|=42,|𝐴|=|𝐴|.οŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺ

All that remains is to calculate (βˆ’1)=βˆ’1,(βˆ’1)=1,(βˆ’1)=βˆ’1,(βˆ’1)=1.οŠͺοŠͺοŠͺοŠͺοŠͺ

We can now derive the determinant of 𝐴 by expanding along column 4. The definition of the determinant gives the formula |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|.οŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺ

Then, the determinant is |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|=(βˆ’1)Γ—(0)Γ—(|𝐴|)+(1)Γ—(βˆ’3)Γ—(39)+(βˆ’1)Γ—(1)Γ—(42)+(1)Γ—(0)Γ—(|𝐴|).οŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺοŠͺ

As we knew, the terms |𝐴|οŠͺ and |𝐴|οŠͺοŠͺ are multiplied by zero, so there was certainly a need to calculate these quantities in this particular example. Simplifying the working above, the answer is |𝐴|=βˆ’159.

In this explainer, we have given only one tool for calculating the determinant of a square matrix, albeit a very powerful tool. One interesting theme emerges when studying the determinant, namely, that we have many choice as to how we calculate it. For the examples of 3Γ—3 and 4Γ—4 matrices that we have given, we had the option to expand along a particular row or column of our choosing. To make the calculations as concise as possible, we looked for the row or column which contained the most number of zeros.

There are many other methods for calculating the determinant of a square matrix, most of which share the common aim of looking for ways to maximize the number of instances of zero that appear in the calculation. One such method uses row operations to reduce a square matrix to an upper-triangular matrix, which has zeroes in every entry under the diagonal entries. In the definition and examples above, we chose to prioritize rows and columns which contained as many zeros as possible, which is evidently a pragmatic approach. This rule of thumb is a key guideline when calculating the determinant of a square matrix, which would be most unwise to ignore!

Key Points

  • The determinant of a 2Γ—2 matrix 𝐴=ο”π‘Žπ‘π‘π‘‘ο  is given by the formula |𝐴|=π‘Žπ‘‘βˆ’π‘π‘.
  • If the matrix 𝐴 has order π‘šΓ—π‘›, then the matrix minor 𝐴 is a matrix of order (π‘šβˆ’1)Γ—(π‘›βˆ’1) that is identical to 𝐴 after removing the 𝑖th row and 𝑗th column.
  • The determinant of an 𝑛×𝑛 matrix 𝐴=[π‘Ž] can be calculated either by expanding along row 𝑖, |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+β‹―+(βˆ’1)π‘Ž|𝐴|,οƒοŠ°οŠ§οƒοŠ§οƒοŠ§οƒοŠ°οŠ¨οƒοŠ¨οƒοŠ¨οƒοŠ°οŠοƒοŠοƒοŠ or by expanding along column 𝑗, |𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+β‹―+(βˆ’1)π‘Ž|𝐴|.οŠ§οŠ°ο…οŠ§ο…οŠ§ο…οŠ¨οŠ°ο…οŠ¨ο…οŠ¨ο…οŠοŠ°ο…οŠο…οŠο…
  • When expanding along a particular row or column, it is wisest to select the row which contains the most number of zeros.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.