Understanding Engineering Mathematics

A 23 =(− 1 )^2 +^3 ∣ ∣ ∣ ∣ a 11 a 12 a 31 a 32 ∣ ∣ ∣ ∣ Then we can easily check that  3 =a 21 A 21 +a 22 A 22 +a 23 A 23 Simila ...

(iii) If two rows or two columns are identical the determinant is zero (follows from (ii)). (iv) Any factors common to a row (co ...

=adeh+bcfg−bceh−adfg =(ad−bc)(eh−fg) ( 41 ➤ ) = ∣ ∣ ∣ ∣ ab cd ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ef gh ∣ ∣ ∣ ∣ as required. Exercises on 13.4 Eval ...

where = ∣ ∣ ∣ ∣ ∣ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ∣ ∣ ∣ ∣ ∣ Again, note the pattern– denominators are always the s ...

= 3 ( 1 + 1 )− 2 (− 2 − 1 ) − 8 = 6 + 6 − 8 =− 3 2 Note that in the case when at least one of the valuesbiis non-zero the system ...

Problem 13.12 Find the determinant and adjoint of A= [− 123 011 − 102 ] and evaluateAAdjA |A|= ∣ ∣ ∣ ∣∣ − 123 011 − 102 ∣ ∣ ∣ ∣∣ ...

Before pressing on to the inverse matrix, let’s think about why we might need it and what it might look like. Remember that we i ...

Problem 13.13 Find, if possible, the inverse of each of the matrices (i)A= [ 1 − 13 221 04 − 5 ] (ii)B= [ 2 − 11 203 1 − 10 ] (i ...

Exercise on 13.6 Where possible, solve the systems of Exercise 13.5 using the inverse matrix. Answer See answer to Exercise 13.5 ...

non-trivial solutionsif|A|=0. All this would be of little interest were it not for the fact that homogeneous systems are very im ...

=λ^2 − 7 λ+ 12 =(λ− 3 )(λ− 4 )= 0 So the eigenvalues areλ=3,λ=4. To find corresponding eigenvectors we must solve the system of ...

In the Applications section (➤403) you are led through an application of eigenvalues and eigenvectors to the simplest vibrating ...

4.If [ cosθ sinθ −sinθ cosθ ] =     √ 3 2 1 2 − 1 2 √ 3 2     determineθin the first quadrant, i.e. 0<θ< 90 °. 5.U ...

9.Write down the following expansions of the determinant |A|= ∣ ∣ ∣ ∣ ∣ 3 − 12 012 4 − 12 ∣ ∣ ∣ ∣ ∣ (i) By first row (ii) By sec ...

(iii) x+y+z=0(iv)x+y+z= 0 x−y+ 2 z= 0 x−y−z= 0 2 x+y− 3 z= 03 x+y+z= 0 (v) 2x− 2 y+z=0(vi)2x−y+z= 0 3 x−y+z= 0 x+y−z= 0 x+y= 03 ...

2.Systems of forces acting on a body in equilibrium can also lead to systems of linear equations. For example, resolution of for ...

We only get non-trivial values for (x 0 ,y 0 ) if the determinant of coefficients is zero: ∣ ∣ ∣ ∣ −λ 1 −ω^2 −λ ∣ ∣ ∣ ∣= ∣ ∣ ∣ ∣ ...

3.a=−1,b=2,c=3,d=2,x=y=3,z= 2 4.θ= 30 ° Additions 2 A= [ 4 − 2 60 ] 2 B= [ 6 − 2 8 ] 2 C=[− 40 ] 2 D= [ 02 − 2 406 − 2 − 64 ] ...

8.(i) 14 (ii) − 6 (iii) − 18 (iv) 4 (v) 0 (vi) 1 (i) 3 ∣ ∣∣ ∣ 12 − 12 ∣ ∣∣ ∣+ ∣ ∣∣ ∣ 02 42 ∣ ∣∣ ∣+^2 ∣ ∣∣ ∣ 01 4 − 1 ∣ ∣∣ ∣ (i ...

(i) 1, [ 1 0 ] ;2, [ 0 1 ] (ii) 3, 1 √ 2 [ 1 1 ] ;4, 1 √ 13 [ 3 2 ] (iii) 1, 1 √ 10 [ − 3 1 ] ;− 1 , 1 √ 2 [ 1 − 1 ] (iv) 4, 1 ...