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Math Algebra (all content) Matrices Determinants & inverses of large matrices

Determinants & inverses of large matrices

    • Determinant of a 3×3 matrix: standard method (1 of 2)
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    • Determinant of a 3×3 matrix: shortcut method (2 of 2)
    • Practice: Determinant of a 3×3 matrix
    • Inverting a 3×3 matrix using Gaussian elimination
    • Inverting a 3×3 matrix using determinants Part 1: Matrix of minors and cofactor matrix
    • Inverting a 3×3 matrix using determinants Part 2: Adjugate matrix
    • Practice: Inverse of a 3×3 matrix
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Math Algebra (all content) Matrices Determinants & inverses of large matrices

Determinant of a 3×3 matrix: standard method (1 of 2)

Determinants & inverses of large matrices

    • Determinant of a 3×3 matrix: standard method (1 of 2)
      This is the currently selected item.

    • Determinant of a 3×3 matrix: shortcut method (2 of 2)
    • Practice: Determinant of a 3×3 matrix
    • Inverting a 3×3 matrix using Gaussian elimination
    • Inverting a 3×3 matrix using determinants Part 1: Matrix of minors and cofactor matrix
    • Inverting a 3×3 matrix using determinants Part 2: Adjugate matrix
    • Practice: Inverse of a 3×3 matrix
    Next tutorial
    Solving equations with inverse matrices
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Determinant of a matrix

Video transcript

As a hint, I will take the
determinant of another 3 by 3 matrix. But it's the exact same
process for the 3 by 3 matrix that you're trying to
find the determinant of. So here is matrix A.
Here, it's these digits. This is a 3 by 3 matrix. And now let's evaluate
its determinant. So what we have to remember
is a checkerboard pattern when we think of
3 by 3 matrices: positive, negative, positive. So first we're going to
take positive 1 times 4. So we could just
write plus 4 times 4, the determinant of 4 submatrix. And when you say,
what's the submatrix? Well, get rid of the column
for that digit, and the row, and then the submatrix
is what's left over. So we'll take the
determinant of its submatrix. So it's 5, 3, 0, 0. Then we move on to the
second item in this row, in this top row. But the checkerboard
pattern says we're going to take
the negative of it. So it's going to be
negative of negative 1– let me do that in a slightly
different color– of negative 1 times the determinant
of its submatrix. You get rid of this
row, and this column. You're left with 4,
3, negative 2, 0. And then finally, you
have positive again. Positive times 1. This 1 right over here. Let me put the positive
in that same blue color. So positive 1, or plus
1 or positive 1 times 1. Really the negative is where
it got a little confusing on this middle term. But positive 1 times 1 times the
determinant of its submatrix. So it's submatrix is
this right over here. You get rid of the row,
get rid of the column 4, 5, negative 2, 0. So now we just have to evaluate
these 2 by 2 determinants. So the determinant
right over here is going to be 5 times
0 minus 3 times 0. And all of that is going
to be multiplied times 4. Well this is going
to be 0 minus 0. So this is all just a 0. So 4 times 0 is just a 0. So this all simplifies to 0. Now let's do this term. We get negative negative 1. So that's positive 1. So let me just make
these positive. Positive 1, or we
could just write plus. Let me just write it here. So positive 1 times
4 times 0 is 0. So 4 times 0 minus
3 times negative 2. 3 times negative
2 is negative 6. So you have 4– oh, sorry,
you have 0 minus negative 6, which is positive 6. Positive 6 times 1 is just 6. So you have plus 6. And then finally you have
this last determinant. You have– so it's going
to be plus 1 times 4 times 0 minus 5 times negative 2. So this is going to be
equal to– it's just going to be equal with–
1 times anything is just the same thing. 4 times 0 is 0. And then 5 times negative
2 is negative 10. But we're going to
subtract a negative 10. So you get positive 10. So this just simplifies
to 10, positive 10. So you're left with,
let me be clear. This is 0, all of this
simplifies to plus 6, and all of this
simplifies to plus 10. And so you are left with, if
you add these up, 6 plus 10 is equal to 16. So the trick here
is to just make sure you remember the
checkerboard pattern, and you don't mess up with all
the negative numbers and all of the multiplying.
Determinant of a 3×3 matrix: shortcut method (2 of 2)
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Determinant of a 3×3 matrix: shortcut method (2 of 2)





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Finding the Determinant of a 3×3 Matrix

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What is a determinant and how do you find it? This lesson explains what a determinant is and shows you a step-by-step process for finding the determinant of a 3 x 3 matrix.

Setting Up the Problem

We are about to go over how to find the determinant for a 3 x 3 matrix, but first we’ll need to know what a determinant is. A determinant is a single specific number associated with a specific square matrix. We should note that determinants are only defined for square matrices. Let’s take a look at the process used to find the determinant for a specific matrix.

  • Step 1 – Write the matrix

We have to know what we’re working on, right? Well, here is the 3 x 3 matrix we are going to be using for this exercise.

3 x 3 Matrix
matrix

The 3 x 3 refers to the number of rows and columns in our matrix. Since it has three rows and three columns, we call it a 3 x 3 matrix. Since the number of columns and rows are equal, this is a square matrix – which means that it will have a determinant.

  • Step 2 – Write the matrix with determinant symbols
Determinant of a 3 x 3 Matrix
matrix determinant

There is only a small difference in this image and the last one: the brackets have turned into straight lines. Mathematically speaking, however, this indicates a very large difference. The matrix represents a whole series of relationships between numbers while the determinant is just a single number.

  • Step 3 – Write the matrix without brackets or determinant symbols

Now that we know the matrix we are working on, what a determinant is and how it’s written – we can start the process of finding the determinant. This step involves just writing the columns and numbers without any other symbols.

3 Columns of 3 x 3 matrix
matrix determinant

Simple, right? Now let’s keep going to the next step.

  • Step 4 – Add the first two columns to the right

Now, to the right of our 3 columns we are going to add two more columns. Not just any columns though – we are simply going to repeat the first two columns from our matrix.

3 Columns of 3 x 3 matrix, plus first two columns repeated
matrix determinant

The dotted line in this picture is just for demonstration purposes – it’s not necessary to put this in when you are working on other determinant problems. Although if it helps you keep track of where you are in the process, you can certainly keep it.

  • Step 5 – Add multiplications of first down diagonal

Look at the image below before we talk any further about this step.

First Down Diagonal Multiplication
matrix determinant

Start with the number in the first row and first column and multiply together the three numbers in the diagonal going down and to the right. In the image above these three numbers are circled. We are going to add the numbers from the down diagonals together.

  • Step 6 – Add multiplications of second and third down diagonals

Repeat Step 5 for the second and third down diagonals. Again, we are choosing the three numbers from our extended matrix that are on a diagonal that goes down and to the right. Once we have these numbers, we are going to multiply them together and add them to our growing expression. Don’t multiply the numbers at this point – we’ll do that later on.

Second and Third Down Diagonal Multiplication
matrix determinant
  • Step 7 – Subtract multiplications of up diagonals

Now we are going to do a similar process with the up diagonals. For each diagonal, we are still going to be choosing three numbers to multiply, but for the up diagonals, we are going to be subtracting the multiplication terms instead of adding.

Up Diagonal Multiplication
matrix determinant

Do you notice how we have a large subtraction sign before each of the up diagonal terms? It’s very easy to get the wrong sign in this step, so make sure you know why it’s there.

  • Step 8 – Compute results

Now that we have all of the terms for computing our determinant, we can start doing the operations. Remember, a negative times a negative is a positive, and if any of the multipliers are 0, then that term is going to be equal to 0.

What we have after all of the above steps is:

Determinant of A

= +(1)(3)(2) + (-4)(-1)(2) + (0)(0)(0) – (2)(3)(0) – (0)(-1)(1) – (2)(0)(-4)

When we evaluate each of the multiplication terms we get:

= 6 + 8 + 0 – 0 – 0 – 0

Solution

Completing the last step of our process will give an answer of 14. This means that the determinant of our matrix turns out to be 14.

How are Determinants Used

Getting this determinant may seem anticlimactic because we don’t yet know how to use it. Determinants are very useful for more advanced math: eigenvalues and eigenvector problems, for example. However, those concepts are beyond what we can cover in this lesson.

You may have noticed that the symbol for the determinant of a matrix looks a lot like the absolute value symbol and we ended up with a positive determinant. Does this mean that determinants can’t be negative?

This seems like a reasonable assumption, but it turns out to be incorrect. To prove it, all we have to do is take our existing matrix and change the sign of every term in it. When we then go through and evaluate the determinant, we will get negative 14 instead positive 14. So, determinants can be both positive and negative.

What about zero determinants? Determinants can be equal to zero, but this only happens when the lines of the square matrix are dependent on each other. Let’s say that our matrix represented a series of 3 equations for 3 unknown variables and we wanted to find out what those variable are. A determinant of 14, because it is not equal to zero, indicates that the three equations are independent of each other – which in turn means that our system of equations has a singular solution. If our determinant was zero, it would essentially mean that we have less than three independent equations, and our system of equations would not have a solution other than all variables equal to zero.


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U.S. Department of Education

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