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Finding the determinant of a 3x3 matrix method 2 Video Lecture - Engineering Mathematics
FAQs on Finding the determinant of a 3x3 matrix method 2 Video Lecture - Engineering Mathematics
| 1. What is the formula for finding the determinant of a 3x3 matrix? |  |
Ans. The formula for finding the determinant of a 3x3 matrix is given by the following expression:
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
where A is the 3x3 matrix with elements a, b, c, d, e, f, g, h, and i.
| 2. Can the determinant of a 3x3 matrix be negative? | |
Ans. Yes, the determinant of a 3x3 matrix can be negative. The sign of the determinant depends on the order of the elements in the matrix and the operations performed during calculation. So, the determinant can be positive, negative, or even zero.
| 3. What is the significance of the determinant in engineering mathematics? | |
Ans. The determinant plays a crucial role in engineering mathematics as it provides important information about the matrix and its properties. It is used to determine if a matrix is invertible, to solve systems of linear equations, calculate areas and volumes, and to find eigenvectors and eigenvalues in various engineering applications.
| 4. Are there any shortcuts or tricks to find the determinant of a 3x3 matrix? | |
Ans. Yes, there is a shortcut method called the "Sarrus' Rule" to find the determinant of a 3x3 matrix. This method involves writing the matrix elements in a specific pattern and performing simple multiplications and additions. However, it is recommended to understand the underlying concept and formula for finding determinants to ensure accuracy and reliability.
| 5. Can the determinant of a 3x3 matrix be zero? | |
Ans. Yes, the determinant of a 3x3 matrix can be zero. When the determinant is zero, it indicates that the matrix is singular (non-invertible) and the system of equations represented by the matrix has no unique solution. This condition is often encountered in engineering problems, and it signifies a special case that requires further analysis or alternative methods for solving the equations.
Text Transcript from Video
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 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.
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Dec 12, 2024 Last updated |
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