Linear equations
Case study Steady-state reactor
Simple example Accumulation = inputs - outputs Steady state inputs = outputs
Simple example Accumulation = inputs - outputs Steady state inputs = outputs
Simple example Accumulation = inputs - outputs 𝑄 1 =2 𝑚 3 𝑚𝑖𝑛 𝑐 1 =25 𝑚𝑔/ 𝑚 3 Flow 1 Flow 3 Flow 2 𝑄 3 =3.5 𝑚 3 𝑚𝑖𝑛 𝑐 2 =? 𝑄 2 =1.5 𝑚 3 𝑚𝑖𝑛 𝑐 2 =10 𝑚𝑔/ 𝑚 3 Steady state inputs = outputs
How to solve that problem? inputs = outputs So? 𝑄 1 𝑐 1 + 𝑄 2 𝑐 2 = 𝑄 3 𝑐 3 50+15=3.5 𝑐 3 Linear equation 𝑐 3 =18.6 𝑚𝑔/ 𝑚 3
More complicated problem Accumulation = inputs - outputs 𝑄 55 =2 𝑚 3 𝑚𝑖𝑛 inputs outputs 𝑄 15 =3 𝑚 3 𝑚𝑖𝑛 outputs 𝑐 5 𝑄 54 =2 𝑚 3 𝑚𝑖𝑛 outputs outputs 𝑄 25 =1 𝑚 3 𝑚𝑖𝑛 𝑄 44 =11 𝑚 3 𝑚𝑖𝑛 inputs outputs inputs outputs inputs outputs 𝑄 01 =5 𝑚 3 𝑚𝑖𝑛 𝑐 01 =20 𝑚𝑔/ 𝑚 3 𝑄 12 =3 𝑚 3 𝑚𝑖𝑛 𝑐 1 𝑐 2 𝑐 4 outputs 𝑄 23 =1 𝑚 3 𝑚𝑖𝑛 outputs 𝑄 31 =1 𝑚 3 𝑚𝑖𝑛 inputs outputs inputs 𝑄 34 =8 𝑚 3 𝑚𝑖𝑛 𝑄 03 =8 𝑚 3 𝑚𝑖𝑛 𝑐 03 =20 𝑚𝑔/ 𝑚 3 𝑐 3
More complicated problem Accumulation = inputs - outputs
How to solve that problem? inputs = outputs So? 𝑄 15 =3 𝑚 3 𝑚𝑖𝑛 outputs 6 𝑐 1 − 𝑐 3 =50 −3 𝑐 1 +3 𝑐 2 =0 − 𝑐 2 +9 𝑐 3 =160 − 𝑐 2 −8 𝑐 3 +11 𝑐 4 −2 𝑐 5 =0 − 3𝑐 1 − 𝑐 2 +4 𝑐 5 =0 inputs outputs 𝑄 01 =5 𝑚 3 𝑚𝑖𝑛 𝑐 01 =20 𝑚𝑔/ 𝑚 3 𝑄 12 =3 𝑚 3 𝑚𝑖𝑛 𝑐 1 Simple? No! outputs 𝑄 31 =1 𝑚 3 𝑚𝑖𝑛
Equations – simple examples How do solve equations? 𝟑𝒙=𝟏
Equations – simple examples How do solve equations? 𝟑𝒙=𝟏 𝟑𝒙 𝟑 = 𝟏 𝟑
Equations – simple examples How do solve equations? 𝟑𝒙=𝟏 𝟑𝒙 𝟑 = 𝟏 𝟑 𝒙= 𝟏 𝟑
Equations – simple examples How do solve equations? 𝒙 𝟐 =𝟒
Equations – simple examples How do solve equations? 𝒙 𝟐 =𝟒 𝒙 𝟐 = 𝟒
Equations – simple examples How do solve equations? 𝒙 𝟐 =𝟒 𝒙 𝟐 = 𝟒 𝒙=𝟐, 𝒙=−𝟐
Equations – simple examples How do solve equations? 𝒆 𝒙 =𝟖
Equations – simple examples How do solve equations? 𝒆 𝒙 =𝟖 𝒍𝒏 𝒆 𝒙 =𝒍𝒏 𝟖
Equations – simple examples How do solve equations? 𝒆 𝒙 =𝟖 𝒍𝒏 𝒆 𝒙 =𝒍𝒏 𝟖 𝒙=𝒍𝒏 𝟖
Equations – simple examples How do solve equations? 𝒅 𝒅𝒙 𝒇(𝒙)=𝟑
Equations – simple examples How do solve equations? 𝒅 𝒅𝒙 𝒇(𝒙)=𝟑 ∫ 𝒅𝒇(𝒙) 𝒅𝒙 𝒅𝒙=∫𝟑𝒅𝒙
Equations – simple examples How do solve equations? 𝒅 𝒅𝒙 𝒇(𝒙)=𝟑 ∫ 𝒅𝒇(𝒙) 𝒅𝒙 𝒅𝒙=∫𝟑𝒅𝒙 ∫𝒅𝒇=𝒇=∫𝟑𝒅𝒙=𝟑𝒙+𝑪
Equations – simple examples How do solve equations? For each type of equation do we have to know special solving procedure?
Equations – simple examples How do solve equations? No!
General equation may be written as: 𝑾𝒉𝒂𝒕 𝒊𝒔 𝒊𝒏 𝒄𝒐𝒎𝒎𝒐𝒏? General equation may be written as: 𝑨𝒙=𝒃
General equation may be written as: 𝑾𝒉𝒂𝒕 𝒊𝒔 𝒊𝒏 𝒄𝒐𝒎𝒎𝒐𝒏? General equation may be written as: 𝑨𝒙=𝒃 unknown
General equation may be written as: 𝑾𝒉𝒂𝒕 𝒊𝒔 𝒊𝒏 𝒄𝒐𝒎𝒎𝒐𝒏? General equation may be written as: 𝑨𝒙=𝒃 𝑨=𝟑∗, 𝟐 , 𝒆 , 𝒅 𝒅𝒙 operation unknown
𝑯𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒊𝒕?
𝑯𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒊𝒕? Same scheme as previously… We use inverse operation to extract x
Same scheme as previously… We use inverse operation to extract x 𝑯𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒊𝒕? Same scheme as previously… We use inverse operation to extract x 𝑨 𝑨 −𝟏 =𝑰
Same scheme as previously… We use inverse operation to extract x 𝑯𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒊𝒕? Same scheme as previously… We use inverse operation to extract x 𝑨 𝑨 −𝟏 =𝑰 operation Inverse operation
Same scheme as previously… We use inverse operation to extract x 𝑯𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒊𝒕? Same scheme as previously… We use inverse operation to extract x 𝑨 𝑨 −𝟏 =𝑰 operation Inverse operation Unit operation
𝑼𝒏𝒊𝒕 𝒐𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒅𝒐𝒆𝒔… Nothing 𝑰𝒙=𝒙𝑰=𝒙
𝑺𝒐 𝒉𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒂𝒏𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏? 𝑨𝒙=𝒃
Simple – apply inverse operation 𝑺𝒐 𝒉𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒂𝒏𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏? Simple – apply inverse operation 𝑨𝒙=𝒃 𝑨𝒙=𝒃 | 𝑨 −𝟏
Simple – apply inverse operation 𝑺𝒐 𝒉𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒂𝒏𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏? Simple – apply inverse operation 𝑨𝒙=𝒃 𝑨𝒙=𝒃 | 𝑨 −𝟏 𝑨 −𝟏 𝑨𝒙= 𝑨 −𝟏 𝒃
Simple – apply inverse operation 𝑺𝒐 𝒉𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒂𝒏𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏? Simple – apply inverse operation 𝑨𝒙=𝒃 𝑨𝒙=𝒃 | 𝑨 −𝟏 𝑨 −𝟏 𝑨𝒙= 𝑨 −𝟏 𝒃 𝑰𝒙= 𝑨 −𝟏 𝒃
Simple – apply inverse operation 𝑺𝒐 𝒉𝒐𝒘 𝒕𝒐 𝒔𝒐𝒍𝒗𝒆 𝒂𝒏𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏? Simple – apply inverse operation 𝑨𝒙=𝒃 𝑨𝒙=𝒃 | 𝑨 −𝟏 𝑨 −𝟏 𝑨𝒙= 𝑨 −𝟏 𝒃 𝑰𝒙= 𝑨 −𝟏 𝒃 𝒙= 𝑨 −𝟏 𝒃
𝑾𝒉𝒂 𝒕 ′ 𝒔 𝒕𝒉𝒆 𝒈𝒂𝒊𝒏?
𝑾𝒉𝒂 𝒕 ′ 𝒔 𝒕𝒉𝒆 𝒈𝒂𝒊𝒏? We have general recipe to solve any equation!
Write down the equation. 𝑾𝒉𝒂 𝒕 ′ 𝒔 𝒕𝒉𝒆 𝒈𝒂𝒊𝒏? We have general recipe to solve any equation! Write down the equation.
Write down the equation. Identify operation acting on x. 𝑾𝒉𝒂 𝒕 ′ 𝒔 𝒕𝒉𝒆 𝒈𝒂𝒊𝒏? We have general recipe to solve any equation! Write down the equation. Identify operation acting on x.
𝑾𝒉𝒂 𝒕 ′ 𝒔 𝒕𝒉𝒆 𝒈𝒂𝒊𝒏? We have general recipe to solve any equation! Write down the equation. Identify operation acting on x. Find inverse operation.
𝑾𝒉𝒂 𝒕 ′ 𝒔 𝒕𝒉𝒆 𝒈𝒂𝒊𝒏? We have general recipe to solve any equation! Write down the equation. Identify operation acting on x. Find inverse operation. Apply it on both sides.
𝑾𝒉𝒂 𝒕 ′ 𝒔 𝒕𝒉𝒆 𝒈𝒂𝒊𝒏? We have general recipe to solve any equation! Write down the equation. Identify operation acting on x. Find inverse operation. Apply it on both sides. Enjoy the result!
𝑾𝒉𝒂𝒕 𝒂𝒃𝒐𝒖𝒕 𝒍𝒊𝒏𝒆𝒂𝒓 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔? Example: 𝒙+𝒚=𝟒 𝒙−𝒚=𝟐
𝑾𝒉𝒂𝒕 𝒂𝒃𝒐𝒖𝒕 𝒍𝒊𝒏𝒆𝒂𝒓 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔? Example: 𝒙+𝒚=𝟒 𝒙−𝒚=𝟐 → 𝒙=𝟑 𝒚=𝟏
𝑯𝒐𝒘 𝒂𝒃𝒐𝒖𝒕 𝒕𝒉𝒊𝒔 𝒐𝒏𝒆? 𝟑𝒙+𝟐𝒚−𝒛+𝒌=𝟑 −𝟐𝟑𝒙−𝟏𝟒𝟓𝒚+𝟏𝟐𝒛−𝟖𝒌=𝝅 𝟓𝒙+𝟕𝒚−𝟐𝟎𝒛−𝟗𝒌=−𝟗𝟐 𝒙+𝒚+𝒛=𝟐
𝑴𝒂𝒕𝒓𝒊𝒄𝒆𝒔 𝑨= 𝟐 𝟏 𝟑 𝟐 𝑩= 𝟏 𝟐 , 𝑪= 𝟏 𝟐 , 𝑫= 𝟒 𝟔 𝟗 𝟖 𝟑 𝟓
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Additon 𝑨= 𝟐 𝟏 𝟑 𝟐 𝑩= 𝟎 𝟐 𝟔 𝟓
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Additon 𝑨= 𝟐 𝟏 𝟑 𝟐 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨+𝑩= 𝟐+𝟎 𝟏+𝟐 𝟑+𝟔 𝟐+𝟓
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Additon 𝑨= 𝟐 𝟏 𝟑 𝟐 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨+𝑩= 𝟐+𝟎 𝟏+𝟐 𝟑+𝟔 𝟐+𝟓
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Additon 𝑨= 𝟐 𝟏 𝟑 𝟐 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨+𝑩= 𝟐+𝟎 𝟏+𝟐 𝟑+𝟔 𝟐+𝟓
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Additon 𝑨= 𝟐 𝟏 𝟑 𝟐 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨+𝑩= 𝟐+𝟎 𝟏+𝟐 𝟑+𝟔 𝟐+𝟓 = 𝟐 𝟑 𝟗 𝟕
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Multiplication 𝑨= 𝟐 𝟏 𝟑 𝟐 , 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨∗𝑩= =
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Multiplication 𝑨= 𝟐 𝟏 𝟑 𝟐 , 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨∗𝑩= 𝟐∗𝟎+𝟏∗𝟔 = 𝟔
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Multiplication 𝑨= 𝟐 𝟏 𝟑 𝟐 , 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨∗𝑩= 𝟐∗𝟎+𝟏∗𝟔 𝟐∗𝟐+𝟏∗𝟓 = 𝟔 𝟗
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Multiplication 𝑨= 𝟐 𝟏 𝟑 𝟐 , 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨∗𝑩= 𝟐∗𝟎+𝟏∗𝟔 𝟐∗𝟐+𝟏∗𝟓 𝟑∗𝟎+𝟐∗𝟔 = 𝟔 𝟗 𝟏𝟐
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Multiplication 𝑨= 𝟐 𝟏 𝟑 𝟐 , 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨∗𝑩= 𝟐∗𝟎+𝟏∗𝟔 𝟐∗𝟐+𝟏∗𝟓 𝟑∗𝟎+𝟐∗𝟔 𝟑∗𝟐+𝟐∗𝟓 = 𝟔 𝟗 𝟏𝟐 𝟏𝟔
𝑾𝒉𝒂𝒕 𝒄𝒂𝒏 𝒘𝒆 𝒅𝒐 𝒘𝒊𝒕𝒉 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔? Division??? 𝑨= 𝟐 𝟏 𝟑 𝟐 , 𝑩= 𝟎 𝟐 𝟔 𝟓 𝑨 𝑩 =???
𝑰𝒏𝒗𝒆𝒓𝒔𝒆 𝒎𝒂𝒕𝒓𝒊𝒙 𝑨 𝑨 −𝟏 = 𝑨 −𝟏 𝑨=𝑰
𝑰𝒏𝒗𝒆𝒓𝒔𝒆 𝒎𝒂𝒕𝒓𝒊𝒙 𝑨 𝑨 −𝟏 = 𝑨 −𝟏 𝑨=𝑰 𝑰= 𝟏 𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟏
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝑨 −𝟏 ? 100% algorithm 1. Compute det(A).
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟏 𝟏 −𝟏 =
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟏 𝟏 −𝟏 = 𝟏∗ −𝟏
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟏 𝟏 −𝟏 = 𝟏∗ −𝟏 −𝟏∗𝟏=−𝟐
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟐 𝟑 𝟎 𝟐 𝟏 𝟓 𝟏 𝟗
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟐 𝟑 𝟎 𝟐 𝟏 𝟓 𝟏 𝟗 𝟏
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟐 𝟑 𝟎 𝟐 𝟏 𝟓 𝟏 𝟗 𝟏∗ −𝟏 𝟏+𝟏 1st 1st
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟐 𝟑 𝟎 𝟐 𝟏 𝟓 𝟏 𝟗 𝟏∗ −𝟏 𝟏+𝟏 ∗ 𝟐 𝟏 𝟏 𝟗 1st 1st
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟐 𝟑 𝟎 𝟐 𝟏 𝟓 𝟏 𝟗 𝟏∗ −𝟏 𝟏+𝟏 ∗ 𝟐 𝟏 𝟏 𝟗 +𝟎∗ −𝟏 𝟐+𝟏 𝟐 𝟑 𝟏 𝟗 𝟏∗ −𝟏 𝟏+𝟏 ∗ 𝟐 𝟏 𝟏 𝟗 +𝟎∗ −𝟏 𝟐+𝟏 𝟐 𝟑 𝟏 𝟗 Type equation here. 1st 2nd
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟐 𝟑 𝟎 𝟐 𝟏 𝟓 𝟏 𝟗 𝟏∗ −𝟏 𝟏+𝟏 ∗ 𝟐 𝟏 𝟏 𝟗 +𝟎∗ −𝟏 𝟐+𝟏 𝟐 𝟑 𝟏 𝟗 +𝟓∗ −𝟏 𝟑+𝟏 ∗ 𝟐 𝟑 𝟐 𝟏 = 1st 3rd
𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕𝒔 𝟏 𝟐 𝟑 𝟎 𝟐 𝟏 𝟓 𝟏 𝟗 𝟏∗ −𝟏 𝟏+𝟏 ∗ 𝟐 𝟏 𝟏 𝟗 +𝟎∗ −𝟏 𝟐+𝟏 𝟐 𝟑 𝟏 𝟗 +𝟓∗ −𝟏 𝟑+𝟏 ∗ 𝟐 𝟑 𝟐 𝟏 = 𝟏∗𝟏∗𝟏𝟕+𝟎+𝟐∗𝟏∗𝟔=𝟓
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝑨 −𝟏 ? 100% algorithm Compute det(A). 𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝑨 −𝟏 ? 100% algorithm Compute det(A). 𝑨 −𝟏 = 𝟏 𝒅𝒆𝒕 𝑨 ∗𝒔𝒕𝒉
𝑾𝒉𝒚 𝒅𝒐 𝒘𝒆 𝒏𝒆𝒆𝒅 𝒊𝒕? 𝒙+𝒚=𝟒 𝒙−𝒚=𝟐 𝟏∗𝒙+𝟏∗𝒚=𝟒 𝟏∗𝒙−𝟏∗𝒚=𝟐
𝑾𝒉𝒚 𝒅𝒐 𝒘𝒆 𝒏𝒆𝒆𝒅 𝒊𝒕? 𝒙+𝒚=𝟒 𝒙−𝒚=𝟐 𝟏 𝟏 𝟏 −𝟏 𝒙 𝒚 = 𝟒 𝟐
𝑾𝒉𝒚 𝒅𝒐 𝒘𝒆 𝒏𝒆𝒆𝒅 𝒊𝒕? 𝒙+𝒚=𝟒 𝒙−𝒚=𝟐 𝑨 𝒙 =𝑏 𝑨 −𝟏 = 𝟏 𝟏 𝟏 −𝟏 −𝟏 𝒃= 𝟒 𝟐
𝑾𝒉𝒚 𝒅𝒐 𝒘𝒆 𝒏𝒆𝒆𝒅 𝒊𝒕? 𝒙+𝒚=𝟒 𝒙−𝒚=𝟐 𝑨 𝒙 =𝑏 𝒙 = 𝑨 −𝟏 𝒃 𝒙+𝒚=𝟒 𝒙−𝒚=𝟐 𝑨 𝒙 =𝑏 𝒙 = 𝑨 −𝟏 𝒃 𝑨 −𝟏 = 𝟏 𝟏 𝟏 −𝟏 −𝟏 𝒃= 𝟒 𝟐
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? 100% algorithm 𝟏 𝟏 𝟏 −𝟏 →
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? Calculate minors 𝟏 𝟏 𝟏 −𝟏 → (−𝟏) 𝟏+𝟏 −𝟏 1st 1st
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? Calculate minors 𝟏 𝟏 𝟏 −𝟏 → (−𝟏) 𝟏+𝟏 −𝟏 (−𝟏) 𝟐+𝟏 ∗|𝟏| 1st 2nd
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? Calculate minors 𝟏 𝟏 𝟏 −𝟏 → (−𝟏) 𝟏+𝟏 −𝟏 (−𝟏) 𝟏+𝟐 ∗ 𝟏 (−𝟏) 𝟐+𝟏 ∗|𝟏| 1st 2nd
𝟏 𝟏 𝟏 −𝟏 → (−𝟏) 𝟏+𝟏 −𝟏 (−𝟏) 𝟏+𝟐 ∗ 𝟏 (−𝟏) 𝟐+𝟏 ∗|𝟏| (−𝟏) 𝟐+𝟐 ∗ 𝟏 𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? Calculate minors 𝟏 𝟏 𝟏 −𝟏 → (−𝟏) 𝟏+𝟏 −𝟏 (−𝟏) 𝟏+𝟐 ∗ 𝟏 (−𝟏) 𝟐+𝟏 ∗|𝟏| (−𝟏) 𝟐+𝟐 ∗ 𝟏 −𝟏 −𝟏 −𝟏 𝟏 𝑻 = −𝟏 −𝟏 −𝟏 𝟏 2nd 2nd
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? Transpose −𝟏 −𝟏 −𝟏 𝟏 𝑻
Transpose
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? Transpose −𝟏 −𝟏 −𝟏 𝟏 𝑻 = −𝟏 −𝟏 −𝟏 𝟏
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝑨 −𝟏 ? 100% algorithm Compute det(A). 𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝑨 −𝟏 ? 100% algorithm Compute det(A). 𝑨 −𝟏 = 𝟏 𝒅𝒆𝒕 𝑨 ∗𝒔𝒕𝒉= =− 𝟏 𝟐 −𝟏 −𝟏 −𝟏 𝟏 = 𝟎.𝟓 𝟎.𝟓 𝟎.𝟓 −𝟎.𝟓
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝑨 −𝟏 ? 100% algorithm Compute det(A). 𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝑨 −𝟏 ? 100% algorithm Compute det(A). 𝑨 −𝟏 = 𝟏 𝒅𝒆𝒕 𝑨 ∗𝒔𝒕𝒉 𝑪𝒐𝒎𝒑𝒖𝒕𝒆 𝒙 = 𝑨 −𝟏 𝒃
𝑯𝒐𝒘 𝒕𝒐 𝒄𝒐𝒎𝒑𝒖𝒕𝒆 𝒔𝒕𝒉? 100% algorithm 𝒙 = 𝑨 −𝟏 𝒃 𝒙 = 𝟎.𝟓 𝟎.𝟓 𝟎.𝟓 −𝟎.𝟓 𝟒 𝟐 = 𝟎.𝟓∗𝟒+𝟎.𝟓∗𝟐 𝟎.𝟓∗𝟒−𝟎.𝟓∗𝟐 = 𝟑 𝟏
𝑬𝒂𝒔𝒚?
𝑬𝒂𝒔𝒚? No!
𝑬𝒂𝒔𝒚? No! 𝑨𝒅𝒗𝒂𝒏𝒕𝒂𝒈𝒆𝒔: Always works – strict , clear procedure, no thinking, no guessing.
𝑬𝒂𝒔𝒚? No! 𝑨𝒅𝒗𝒂𝒏𝒕𝒂𝒈𝒆𝒔: Always works – strict , clear procedure, no thinking, no guessing. Test for solubility : det(A) ≠0.
𝑬𝒂𝒔𝒚? No! 𝑨𝒅𝒗𝒂𝒏𝒕𝒂𝒈𝒆𝒔: Always works – strict , clear procedure, no thinking, no guessing. Test for solubility : det(A) ≠0. Test for possible „insane” results: det(A) ≈0.
𝑬𝒂𝒔𝒚? No! 𝑨𝒅𝒗𝒂𝒏𝒕𝒂𝒈𝒆𝒔: Always works – strict , clear procedure, no thinking, no guessing. Test for solubility : det(A) ≠0. Test for possible „insane” results: det(A) ≈0. Works perfectly with a computer.
𝑬𝒂𝒔𝒚? No! 𝑨𝒅𝒗𝒂𝒏𝒕𝒂𝒈𝒆𝒔: Always works – strict , clear procedure, no thinking, no guessing. Test for solubility : det(A) ≠0. Test for possible „insane” results: det(A) ≈0. Works perfectly with a computer.