Linear Algebra-Determinants An Overview- 1, guaranteed and verified 100% Pass

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Determinants: An Overview


For each 𝑛 × 𝑛 matrix, 𝐴, we can calculate a number called the determinant of 𝐴,
𝑑𝑒𝑡(𝐴). This is often written as |𝐴|.
Case 1. 1 × 1 matrices. If 𝐴 = [𝑎] then 𝑑𝑒𝑡(𝐴) = |𝐴| = 𝑎.


𝑎11 𝑎12
Case 2. 2 × 2 matrices. Let 𝐴 = [𝑎 𝑎22 ], the we have:
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det(𝐴) = |𝐴| = 𝑎11 𝑎22 − 𝑎21 𝑎12 .


3 −2
Ex. | | = 3(5) − (4)(−2) = 15 + 8 = 23.
4 5


Def. Let 𝐴 = [𝑎𝑖𝑗 ] be an 𝑛 × 𝑛 matrix and let 𝑀𝑖𝑗 be the
(𝑛 − 1) × (𝑛 − 1) matrix obtained from deleting the row and column containing
𝑎𝑖𝑗. The determinant of (𝑀𝑖𝑗 ) is called the Minor of 𝑎𝑖𝑗 . We define the Cofactor,
𝐴𝑖𝑗 , of 𝑎𝑖𝑗 by:

𝐴𝑖𝑗 = (−1)𝑖+𝑗 det(𝑀𝑖𝑗 ).


𝑎11 ⋯ 𝑎1𝑛
Def. The Determinant of an 𝒏 × 𝒏 matrix 𝑨, where 𝐴 = [ ⋮ ⋱ ⋮ ], is given
𝑎𝑛1 ⋯ 𝑎𝑛𝑛
by: 𝐷𝑒𝑡 (𝐴) = 𝑎11 𝐴11 + 𝑎12 𝐴12 + ⋯ + 𝑎1𝑛 𝐴1𝑛
where 𝐴1𝑗 = (−1)1+𝑗 det(𝑀1𝑗 ) ; 𝑗 = 1,2, ⋯ , 𝑛.

, 2


Case 3. 3x3 matrices. Let 𝐴 be:
𝑎11 𝑎12 𝑎13
𝐴 = [𝑎21 𝑎22 𝑎23 ]
𝑎31 𝑎32 𝑎33


Then 𝑑𝑒𝑡(𝐴) = 𝑎11 𝐴11 + 𝑎12 𝐴12 + 𝑎13 𝐴13;

𝑎22 𝑎23 𝑎22 𝑎23
where: 𝐴11 = (−1)1+1 det [𝑎 𝑎33 ] = det [ 𝑎32 𝑎33 ]
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𝑎21 𝑎23 𝑎21 𝑎23
𝐴12 = (−1)1+2 det [𝑎 𝑎33 ] = − det [𝑎31 𝑎33 ]
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𝑎21 𝑎22 𝑎21 𝑎22
𝐴13 = (−1)1+3 det [𝑎 𝑎32 ] = det [ 𝑎31 𝑎32 ].
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2 1 3
Ex. Find 𝑑𝑒𝑡(𝐴) where 𝐴 = [4 1 2].
1 2 3


2 1 3
1 2 4 2 4 1
|4 1 2| = 2 | |− 1| |+ 3| |
2 3 1 3 1 2
1 2 3
= 2[(1)(3) − 2(2)] − 1[4(3) − 1(2)] + 3[4(2) − 1(1)]
= 2(3 − 4) − 1(12 − 2) + 3(8 − 1)
= 2(−1) − 10 + 3(7)
= 9.