Determinant of a Matrix
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Question 1 of 4
1. Question
Find the determinant of the matrix:`[[4,2],[-1,3]]`- (14)
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A determinant is a real number associated with a square matrix.Determinant of a `2xx2` Matrix
If \begin{bmatrix}
\color{#007DDC}{a} & \color{#9a00c7}{b} \\
\color{#9a00c7}{c} & \color{#007DDC}{d}
\end{bmatrix} then $$|A|=\color{#007DDC}{ad}-\color{#9a00c7}{bc}$$First, label the values of the matrix`[[a,b],[c,d]]=[[4,2],[-1,3]]``a=4``c=-1``b=2``d=3`Substitute the values into the Determinant Formula\begin{vmatrix}
\color{#007DDC}{a} & \color{#9a00c7}{b} \\
\color{#9a00c7}{c} & \color{#007DDC}{d}
\end{vmatrix}`=` `ad` `-``bc` Determinant Formula \begin{vmatrix}
\color{#007DDC}{4} & \color{#9a00c7}{2} \\
\color{#9a00c7}{-1} & \color{#007DDC}{3}
\end{vmatrix}`=` `4*3` `-``2*(-1)` Determinant Formula `=` `12-(-2)` `=` `14` `14` -
Question 2 of 4
2. Question
Find the determinant of the matrix:`[[6,-2],[3,-5]]`- (-24)
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A determinant is a real number associated with a square matrix.Determinant of a `2xx2` Matrix
If \begin{bmatrix}
\color{#007DDC}{a} & \color{#9a00c7}{b} \\
\color{#9a00c7}{c} & \color{#007DDC}{d}
\end{bmatrix} then $$|A|=\color{#007DDC}{ad}-\color{#9a00c7}{bc}$$First, label the values of the matrix`[[a,b],[c,d]]=[[6,-2],[3,-5]]``a=6``c=3``b=-2``d=-5`Substitute the values into the Determinant Formula\begin{vmatrix}
\color{#007DDC}{a} & \color{#9a00c7}{b} \\
\color{#9a00c7}{c} & \color{#007DDC}{d}
\end{vmatrix}`=` `ad` `-``bc` Determinant Formula \begin{vmatrix}
\color{#007DDC}{6} & \color{#9a00c7}{-2} \\
\color{#9a00c7}{3} & \color{#007DDC}{-5}
\end{vmatrix}`=` `6*(-5)` `-``-2*3` Determinant Formula `=` `-30-(-6)` `=` `-24` `-24` -
Question 3 of 4
3. Question
Find the determinant of the matrix:`[[3,0],[-4,7]]`- (21)
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A determinant is a real number associated with a square matrix.Determinant of a `2xx2` Matrix
If \begin{bmatrix}
\color{#007DDC}{a} & \color{#9a00c7}{b} \\
\color{#9a00c7}{c} & \color{#007DDC}{d}
\end{bmatrix} then $$|A|=\color{#007DDC}{ad}-\color{#9a00c7}{bc}$$First, label the values of the matrix`[[a,b],[c,d]]=[[3,0],[-4,7]]``a=3``c=-4``b=0``d=7`Substitute the values into the Determinant Formula\begin{vmatrix}
\color{#007DDC}{a} & \color{#9a00c7}{b} \\
\color{#9a00c7}{c} & \color{#007DDC}{d}
\end{vmatrix}`=` `ad` `-``bc` Determinant Formula \begin{vmatrix}
\color{#007DDC}{3} & \color{#9a00c7}{0} \\
\color{#9a00c7}{-4} & \color{#007DDC}{7}
\end{vmatrix}`=` `3*7` `-``0*(-4)` Determinant Formula `=` `21-0` `=` `21` `21` -
Question 4 of 4
4. Question
Find the determinant of the matrix:`A=[[4,1,1],[-2,0,1],[1,-1,2]]`- (11)
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A determinant is a real number associated with a square matrix.Determinant of a `3xx3` Matrix
If $$A=$$\begin{bmatrix}
\color{#D800AD}{a} & \color{#D800AD}{b} & \color{#D800AD}{c} \\
d & e & f \\
g & h & k
\end{bmatrix} then $$|A|=\color{#D800AD}{a}$$\begin{vmatrix}
e & f \\
h & k
\end{vmatrix}$$-\color{#D800AD}{b}$$\begin{vmatrix}
d & f \\
g & k
\end{vmatrix}$$+\color{#D800AD}{c}$$\begin{vmatrix}
d & e \\
g & h
\end{vmatrix}Determinant of a `2xx2` Matrix
If \begin{bmatrix}
\color{#007DDC}{a} & \color{#9a00c7}{b} \\
\color{#9a00c7}{c} & \color{#007DDC}{d}
\end{bmatrix} then $$|A|=\color{#007DDC}{ad}-\color{#9a00c7}{bc}$$First, label the values of the matrix\begin{bmatrix}
\color{#D800AD}{a} & \color{#D800AD}{b} & \color{#D800AD}{c} \\
d & e & f \\
g & h & k
\end{bmatrix}$$=$$\begin{bmatrix}
\color{#D800AD}{4} & \color{#D800AD}{1} & \color{#D800AD}{1} \\
-2 & 0 & 1 \\
1 & -1 & 2
\end{bmatrix}`a=4``d=-2``g=1``b=1``e=0``h=-1``c=-1``f=1``k=2`Substitute the values into the Determinant Formula and solve each term`|A|` `=` \begin{vmatrix}
\color{#D800AD}{a} & \color{#D800AD}{b} & \color{#D800AD}{c} \\
d & e & f \\
g & h & k
\end{vmatrix}`=` $$\color{#D800AD}{a}$$\begin{vmatrix}
e & f \\
h & k
\end{vmatrix}$$-\color{#D800AD}{b}$$\begin{vmatrix}
d & f \\
g & k
\end{vmatrix}$$+\color{#D800AD}{c}$$\begin{vmatrix}
d & e \\
g & h
\end{vmatrix}Determinant of a `3xx3` Matrix \begin{vmatrix}
\color{#D800AD}{4} & \color{#D800AD}{1} & \color{#D800AD}{1} \\
-2 & 0 & 1 \\
1 & -1 & 2
\end{vmatrix}`=` $$\color{#D800AD}{4}$$\begin{vmatrix}
\color{#007DDC}{0} & \color{#9a00c7}{1} \\
\color{#9a00c7}{-1} & \color{#007DDC}{2}
\end{vmatrix}$$-\color{#D800AD}{1}$$\begin{vmatrix}
\color{#007DDC}{-2} & \color{#9a00c7}{1} \\
\color{#9a00c7}{1} & \color{#007DDC}{2}
\end{vmatrix}$$+\color{#D800AD}{1}$$\begin{vmatrix}
\color{#007DDC}{-2} & \color{#9a00c7}{0} \\
\color{#9a00c7}{1} & \color{#007DDC}{-1}
\end{vmatrix}Substitute values `=` $$[\color{#D800AD}{4}(\color{#007DDC}{0\cdot2}-\color{#9a00c7}{1\cdot(-1)})]-[\color{#D800AD}{1}(\color{#007DDC}{-2\cdot2}-\color{#9a00c7}{1\cdot1})]+[\color{#D800AD}{1}(\color{#007DDC}{-2\cdot(-1)}-\color{#9a00c7}{0\cdot1})]$$ `=` `4(0-(-1))-1(-4-1)+1(2-0)` `=` `4(1)-1(-5)+1(2)` `=` `4+5+2` `|A|` `=` `11` `11`
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