133 - FML 3 - 18.1W e.g. 71

Basics of Determinants

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Wize University Linear Algebra Textbook > Determinants

(Optional) Shortcut for 3x3 Determinants

3 Activities

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   ⁣B‾=  [1234−112x2]\bcb{\boldsymbol{ \ul{B} =  \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2  \end{bmatrix} }}B​=  ​142​2−1x​312​​ such that  det(  ⁣B‾)=−8 \bcb{\boldsymbol{ \det {\ul{B}} = -8 }}det(B​)=−8  , then x\bcb{\boldsymbol{ x }}x is:
(a)  111(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)111​(b)−1(c)0(d)None of the above

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133 - FML 3 - 18.1W e.g. 17.3

 If   ⁣A‾\bcb{\boldsymbol{ \ul{A} }}A​ is a 3×3\bcb{\boldsymbol{ 3 \times 3}}3×3 matrix with det(  ⁣A‾)=−4\bcb{\boldsymbol{ \det{\ul{A}} = -4}}det(A​)=−4, find:
det(−AT)\boldsymbol{ \det{-\mtran{A}} }det(−AT)

133 - FML 3 - 18.1W e.g. 11

 Find all possible values of det(  ⁣A‾)\bcb{\boldsymbol{ \det{\ul{A}} }}det(A​) given that  adj(A)=  [124−122018]\bcb{\boldsymbol{ adj(A) =  \begin{bmatrix} 1 & 2 & 4 \\ -1 & 2 & 2 \\ 0 & 1 & 8  \end{bmatrix} }}adj(A)=  ​1−10​221​428​​

Basics of Determinants

Answer the following true/false questions.

 Compute the determinant of
[31−111023−20341022]\begin{bmatrix}
3&1&−1&1\\
1&0&2&3\\
−2&0&3&4\\
1&0&2&2

\end{bmatrix}​31−21​1000​−1232​1342​​

Find the determinant of A=[0203−140−13110102−2]A=\begin{bmatrix}
0&2&0&3\\
−1&4&0&−1\\
3&1&1&0\\
1&0&2&−2
\end{bmatrix}A=​0−131​2410​0012​3−10−2​​

Basics of Determinants

Let CCC be an invertible matrix with det(C)=3\det C = 3det(C)=3, and let B=[−2−1001000−5]B=
\left[
\begin{array}{rrr}
-2 & -1 & 0\\
0 & 1 & 0\\
0 & 0 & -5\\
\end{array}
\right]B=​−200​−110​00−5​​. 
If A=C−1BCA = C^{-1}BCA=C−1BC, calculate det(A)\det Adet(A).

Determinant Properties Practice Question

If 𝑨 = 𝑪-1𝟏𝑩𝑪, where C is an invertible matrix, and det(𝑩)=12;det(𝑪)=3.Calculate  det(𝑨).det(𝑩) = 12; det(𝑪) = 3. \text{Calculate}\ \  det(𝑨).det(B)=12;det(C)=3.Calculate  det(A).

133 - FML 1 - 18.1W e.g. 7

Recall that diagonal matrices are substantially easier to work with than non-diagonal matrices:
e.g. If
 A=diag(a,b,c)=[a000b000c]\text{A}=\text{diag}\left(a,b,c\right) = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}A=diag(a,b,c)=​a00​0b0​00c​​ 

Practice: Determinant Property

Practice Question: Determinant Property
Let 𝐶 be an invertible matrix, det (𝐵) = 10, and det (𝐶) = 3. 
If 𝐴 = 𝐶−1𝐵𝐶, calculate det (𝐴).

Determinant Properties Practice Question

If 𝑨 = 𝑪-1𝟏𝑩𝑪, where C is an invertible matrix, and det(𝑩)=12;det(𝑪)=3.Calculate  det(𝑨).det(𝑩) = 12; det(𝑪) = 3. \text{Calculate}\ \  det(𝑨).det(B)=12;det(C)=3.Calculate  det(A).

Basics of Determinants

Find det[32−4101−12−1−750−6−48−2]\text{det}\begin{bmatrix}
3&2&-4&1\\
0&1&-1&2\\
-1&-7&5&0\\
-6&-4&8&-2
\end{bmatrix}det​30−1−6​21−7−4​−4−158​120−2​​.

Determinants: Basics

If A=[1234021−500−30000−1]A=
\begin{bmatrix}
1&2&3&4\\
0&2&1&-5\\
0&0&-3&0\\
0&0&0&-1
\end{bmatrix}A=​1000​2200​31−30​4−50−1​​, calculate det(Adj A)det(Adj\ A)det(Adj A).

Determinants: Basics

AAA is a 3×33\times 33×3 matrix where det A=2det\ A=2det A=2.
Find det(5A2A−1AT)det(5A2A^{-1}A^T)det(5A2A−1AT).

Determinants: Basics

A=[45−12], B=[−10002100302010013−4−10], C=[−155−25], I10=10×10  identity matrixA=
\begin{bmatrix}
4&5\\
-1&2
\end{bmatrix},
\ 
B=
\begin{bmatrix}
-1&0&0&0\\
2&1&0&0\\
30&20&10&0\\
1&3&-4&-10
\end{bmatrix},
\ 
C=
\begin{bmatrix}
-1&5\\
5&-25
\end{bmatrix}, 
\ 
I_{10}=10\times 10\ \text{ identity matrix}A=[4−1​52​], B=​−12301​01203​0010−4​000−10​​, C=[−15​5−25​], I10​=10×10  identity matrix
Compute det(AC)−det(10BT)+det(2I10)det(AC)-det(10B^T)+det(2I_{10})det(AC)−det(10BT)+det(2I10​).

Determinants: Basics

Find the determinant of the matrix [2−41−3]\begin{bmatrix}2&-4\\
1&-3\end{bmatrix}[21​−4−3​].

More (Optional) Shortcut for 3x3 Determinants Questions:

Shortcut for 3x3 Determinants

Let A=[−61aa−2001−2]A=\left[\begin{array}{rrr}
-6&1& a\\
a&-2&0\\
0&1& -2
\end{array}\right]A=​−6a0​1−21​a0−2​​. Find the value(s) of aaa such that AAA is not invertible.

Determinants of Large Matrices

B=[40151002−2]B=
\begin{bmatrix}
4&0&1\\
5&1&0\\
0&2&-2
\end{bmatrix}B=​450​012​10−2​​
Find det(B)\det Bdet(B).

Practice: Determinant of $3\times3$ Matrix

Let A=[61aa−20012]A=\left[\begin{array}{rrr}
6&1&a\\
a&-2&0\\
0&1&2
\end{array}\right]A=​6a0​1−21​a02​​ using cofactor expansion along any row or column
Find the value(s) of aaa that will make AAA invertible

Practice: Determinant

Practice Question: Determinant
Find the determinant of [40151002−2]\begin{bmatrix}
4&0&1\\
5&1&0\\
0&2&-2
\end{bmatrix}​450​012​10−2​​.

Practice: Determinant

Practice: Determinant
Find the determinant of [40151002−2]\begin{bmatrix}
4&0&1\\
5&1&0\\
0&2&-2
\end{bmatrix}​450​012​10−2​​.

133 - FML 3 - 18.1W e.g. 71

Related Topics

Basics of Determinants

(Optional) Shortcut for 3x3 Determinants

$\bcb{\boldsymbol{ \ul{B} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2 \end{bmatrix} }}$ such that $\bcb{\boldsymbol{ \det {\ul{B}} = -8 }}$ , then $\bcb{\boldsymbol{ x }}$ is:

More Basics of Determinants Questions:

133 - FML 3 - 18.1W e.g. 17.3

133 - FML 3 - 18.1W e.g. 11

Basics of Determinants

Basics of Determinants

Determinant Properties Practice Question

133 - FML 1 - 18.1W e.g. 7

Practice: Determinant Property

Determinant Properties Practice Question

Basics of Determinants

Determinants: Basics

Determinants: Basics

Determinants: Basics

Determinants: Basics

More (Optional) Shortcut for 3x3 Determinants Questions:

Shortcut for 3x3 Determinants

Determinants of Large Matrices

Practice: Determinant of $3\times3$ Matrix

Practice: Determinant

Practice: Determinant

133 - FML 3 - 18.1W e.g. 71

Related Topics

Basics of Determinants

(Optional) Shortcut for 3x3 Determinants

⁣B‾= [1234−112x2]\bcb{\boldsymbol{ \ul{B} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2 \end{bmatrix} }}B​= ​142​2−1x​312​​ such that det( ⁣B‾)=−8 \bcb{\boldsymbol{ \det {\ul{B}} = -8 }}det(B​)=−8 , then x\bcb{\boldsymbol{ x }}x is:

More Basics of Determinants Questions:

133 - FML 3 - 18.1W e.g. 17.3

133 - FML 3 - 18.1W e.g. 11

Basics of Determinants

Basics of Determinants

Determinant Properties Practice Question

133 - FML 1 - 18.1W e.g. 7

Practice: Determinant Property

Determinant Properties Practice Question

Basics of Determinants

Determinants: Basics

Determinants: Basics

Determinants: Basics

Determinants: Basics

More (Optional) Shortcut for 3x3 Determinants Questions:

Shortcut for 3x3 Determinants

Determinants of Large Matrices

Practice: Determinant of $3\times3$ Matrix

Practice: Determinant

Practice: Determinant

$\bcb{\boldsymbol{ \ul{B} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2 \end{bmatrix} }}$ such that $\bcb{\boldsymbol{ \det {\ul{B}} = -8 }}$ , then $\bcb{\boldsymbol{ x }}$ is: