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Determinants of Large Matrices
Related Topics
Wize University Linear Algebra Textbook > Determinants
Determinants of Large Matrices
5 Activities
Wize University Linear Algebra Textbook > Determinants
(Optional) Shortcut for 3x3 Determinants
3 Activities
B
=
[
4
0
1
5
1
0
0
2
−
2
]
B= \begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}
B
=
4
5
0
0
1
2
1
0
−
2
Find
det
(
B
)
\det B
det
(
B
)
.
det
(
B
)
=
\det B =
det
(
B
)
=
I don't know
Check Submission
More Determinants of Large Matrices Questions:
133 - FML 3 - 18.1W e.g. 8
Find the determinant of the matrix
A
=
[
1
0
2
−
3
2
1
−
2
1
−
1
0
1
−
1
0
−
2
1
−
1
]
\bcb{\boldsymbol{ A = \begin{bmatrix} 1 & 0 & 2 & -3 \\ 2 & 1 & -2 & 1 \\ -1 & 0 & 1 & -1 \\ 0 & -2 & 1 & - 1\end{bmatrix} }}
A
=
1
2
−
1
0
0
1
0
−
2
2
−
2
1
1
−
3
1
−
1
−
1
133 - FML 3 - 18.1W e.g. 77
Find the determinant of the matrix
A
=
[
1
0
2
−
3
0
2
1
−
2
1
0
−
1
0
1
1
0
0
−
2
1
−
1
1
0
0
1
−
1
0
]
\bcb{\boldsymbol{A = \begin{bmatrix} 1 & 0 & 2 & -3 & 0 \\ 2 & 1 & -2 & 1 & 0 \\ -1 & 0 & 1 & 1 & 0 \\ 0 & -2 & 1 & -1 & 1 \\ 0 & 0 & 1 & -1 & 0 \end{bmatrix} }}
A
=
1
2
−
1
0
0
0
1
0
−
2
0
2
−
2
1
1
1
−
3
1
1
−
1
−
1
0
0
0
1
0
Determinants of Large Matrices
Compute the determinant of
A
=
[
−
2
1
4
−
1
1
0
−
1
2
5
−
1
2
1
0
0
3
−
1
]
A=\left[\begin{array}{rrrr} -2&1&4&-1\\ 1&0&-1&2\\ 5&-1&2&1\\ 0&0&3&-1 \end{array}\right]
A
=
−
2
1
5
0
1
0
−
1
0
4
−
1
2
3
−
1
2
1
−
1
.
Determinants and Inverse
Let
B
=
[
1
−
1
0
2
1
1
1
0
0
0
−
1
1
−
2
1
−
2
−
1
]
B=\left[\begin{array}{rrrr} 1&-1&0&2\\ 1&1&1&0\\ 0&0&-1&1\\ -2&1&-2&-1 \end{array}\right]
B
=
1
1
0
−
2
−
1
1
0
1
0
1
−
1
−
2
2
0
1
−
1
, and note that
det
(
B
)
=
−
1
\text{det}(B)=-1
det
(
B
)
=
−
1
.
Use the inverse formula to find the
(
4
,
3
)
(4,3)
(
4
,
3
)
-entry of
B
−
1
B^{-1}
B
−
1
.
Practice: Determinant
Practice Question: Determinant
Find the determinant of
[
4
0
1
5
1
0
0
2
−
2
]
\begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}
4
5
0
0
1
2
1
0
−
2
.
Practice: Determinant
Practice: Determinant
Find the determinant of
[
4
0
1
5
1
0
0
2
−
2
]
\begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}
4
5
0
0
1
2
1
0
−
2
.
Cramer's Rule: Linear Systems
Let
A
=
[
−
1
−
2
4
3
0
−
2
1
5
−
1
]
A=\begin{bmatrix} -1&-2&4\\ 3&0&-2\\ 1&5&-1 \end{bmatrix}
A
=
−
1
3
1
−
2
0
5
4
−
2
−
1
.
(a) Find
det
A
\text{det}A
det
A
.
(b) User Cramer's Rule to find the value of
x
x
x
in the unique solution to
A
x
=
b
A\bm{x}=\bm{b}
A
x
=
b
, where
x
=
[
x
y
z
]
\bm{x}=\begin{bmatrix} x\\y\\z \end{bmatrix}
x
=
x
y
z
and
b
=
[
0
0
−
2
]
\bm{b}=\begin{bmatrix} 0\\0\\-2 \end{bmatrix}
b
=
0
0
−
2
Determinants and Inverses
Let
A
=
[
a
b
c
d
e
f
1
2
−
3
]
A=\begin{bmatrix} a&b&c\\ d&e&f\\ 1&2&-3 \end{bmatrix}
A
=
a
d
1
b
e
2
c
f
−
3
, where it is known that
det
[
a
b
d
e
]
=
5
\text{det}\begin{bmatrix} a&b\\d&e \end{bmatrix}=5
det
[
a
d
b
e
]
=
5
,
det
[
a
c
d
f
]
=
−
2
\text{det}\begin{bmatrix} a&c\\d&f \end{bmatrix}=-2
det
[
a
d
c
f
]
=
−
2
and
det
[
b
c
e
f
]
=
3
\text{det}\begin{bmatrix} b&c\\e&f \end{bmatrix}=3
det
[
b
e
c
f
]
=
3
.
(a) Find
det
A
\text{det}A
det
A
.
(b) Find the (2,3)-entry of
Adj
A
\text{Adj}A
Adj
A
.
Determinants of Large Matrices
Given
A
=
[
4
5
9
0
9
3
−
2
0
−
1
]
A= \begin{bmatrix} 4&5&9\\ 0&9&3\\ -2&0&-1\\ \end{bmatrix}
A
=
4
0
−
2
5
9
0
9
3
−
1
,
a.) Find the (2, 1)-minor of A.
b.) Find the (1, 3)-cofactor of A.
Determinants of Large Matrices
Compute the determinant of
B
=
[
0
2
1
−
4
−
1
3
2
1
0
0
1
−
2
−
1
3
2
2
]
B= \begin{bmatrix} 0&2&1&-4\\ -1&3&2&1\\ 0&0&1&-2\\ -1&3&2&2 \end{bmatrix}
B
=
0
−
1
0
−
1
2
3
0
3
1
2
1
2
−
4
1
−
2
2
.
More (Optional) Shortcut for 3x3 Determinants Questions:
133 - FML 3 - 18.1W e.g. 71
B
‾
=
[
1
2
3
4
−
1
1
2
x
2
]
\bcb{\boldsymbol{ \ul{B} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2 \end{bmatrix} }}
B
=
1
4
2
2
−
1
x
3
1
2
such that
det
(
B
‾
)
=
−
8
\bcb{\boldsymbol{ \det {\ul{B}} = -8 }}
det
(
B
)
=
−
8
, then
x
\bcb{\boldsymbol{ x }}
x
is:
(a)
1
11
(b)
−
1
(c)
0
(d)
None
of
the
above
\text{(a)}\;\boldsymbol{ \frac{1}{11} } \qquad\qquad\qquad \text{(b)}\; \boldsymbol{ -1}\qquad\qquad\qquad \text{(c)}\; \boldsymbol{ 0}\qquad\qquad\qquad \text{(d)}\; \text{\bf{None of the above}}
(a)
11
1
(b)
−
1
(c)
0
(d)
None of the above
Shortcut for 3x3 Determinants
Let
A
=
[
−
6
1
a
a
−
2
0
0
1
−
2
]
A=\left[\begin{array}{rrr} -6&1& a\\ a&-2&0\\ 0&1& -2 \end{array}\right]
A
=
−
6
a
0
1
−
2
1
a
0
−
2
. Find the value(s) of
a
a
a
such that
A
A
A
is
not invertible
.
Practice: Determinant of $3\times3$ Matrix
Let
A
=
[
6
1
a
a
−
2
0
0
1
2
]
A=\left[\begin{array}{rrr} 6&1&a\\ a&-2&0\\ 0&1&2 \end{array}\right]
A
=
6
a
0
1
−
2
1
a
0
2
using cofactor expansion along any row or column
Find the value(s) of
a
a
a
that will make
A
A
A
invertible
Practice: Determinant
Practice Question: Determinant
Find the determinant of
[
4
0
1
5
1
0
0
2
−
2
]
\begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}
4
5
0
0
1
2
1
0
−
2
.
Practice: Determinant
Practice: Determinant
Find the determinant of
[
4
0
1
5
1
0
0
2
−
2
]
\begin{bmatrix} 4&0&1\\ 5&1&0\\ 0&2&-2 \end{bmatrix}
4
5
0
0
1
2
1
0
−
2
.