Given `A= ((4, 3, 2),(5,6,3),(3,5,2));B = ((a, b, c),(d,e,f),(g,h,i))` Find B so that AB = I = BA as follows: First equate entries on the two sides of the equation AB = I . Then solve the...
Given `A= ((4, 3, 2),(5,6,3),(3,5,2));B = ((a, b, c),(d,e,f),(g,h,i))`
Find B so that AB = I = BA as follows:
First equate entries on the two sides of the equation AB = I . Then solve the resulting nine equations for a, b, c, d, e, f, g, h and i.
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You need to multiply the matrices A and B such that:
`((4,3,2),(5,6,3),(3,5,2))*((a,b,c),(d,e,f),(g,h,i)) = ((1,0,0),(0,1,0),(0,0,1))`
`((4a+3d+2g, 4b+3e+2h, 4c+3f+2i),(5a+6d+3g, 5b+6e+3h, 5c+6f+3i),(3a+5d+2g, 3b+5+2h, 3c+5f+2i)) = ((1,0,0),(0,1,0),(0,0,1))`
Equating corresponding terms yields:
`4a+3d+2g = 1 `
`4b+3e+2h = 0`
`4c+3f+2i = 0 `
`5a+6d+3g = 0 `
`5b+6e+3h = 1 `
`5c+6f+3i = 0 `
`3a+5d+2g = 0 `
`3b+5e+2h = 0 `
`3c+5f+2i = 1`
You need to consider the equations that contain a,d,g such that:
`4a+3d+2g = 1 `
`5a+6d+3g = 0 `
`3a+5d+2g = 0`
You need to subtract the third equation from the first such that:
`a - 2d = 1`
You need to multiply the first equation by 3 and the second equation by -2 and then you should add the new equations such that:
`12a + 9d + 6g = 3`
` -10a - 12d - 6g = 0`
`2a - 3d = 3`
You need to multiply the equation a - 2d = 1 by -2 such that:
`-2a + 4d = -2`
Adding this equation to 2a - 3d = 3 yields:
`2a - 3d - 2a + 4d= 3 - 2`
`d = 1 =gt a - 2 = 1 =gt a = 3`
`15+6+3g = 0 =gt 3g = -21 =gt g = -7`
You need to consider the equations that contain b,e,h such that:
`4b+3e+2h = 0 `
`5b+6e+3h = 1 `
`3b+5e+2h = 0`
Subtracting the third equation from the first yields:
`b - 2e = 0 =gt b = 2e`
Substituting 2e for b in the first and second equations yields:
`11e + 2h = 0 `
`16e + 3h = 1`
You need to multiply by 3 the equation 11e + 2h = 0 and by -2 the equation 16e + 3h = 1 such that:
`33e + 6h = 0 `
`-32e - 6h = -2`
`33e + 6h -32e - 6h = 0-2`
`e = -2 =gt b = -4`
`-16-6+2h = 0 =gt 2h = 22 =gt h = 11`
You need to consider the equations that contain c,f,i such that:
`4c+3f+2i = 0 `
`5c+6f+3i = 0 `
`3c+5f+2i = 1`
`4c+3f+2i - 3c-5f-2i = -1`
` c - 2f = -1 =gt c = 2f - 1`
Substituting 2f - 1 for c in the first and second equations yields:
`4(2f-1)+3f+2i = 0=gt 11f + 2i = 4 `
`5(2f-1)+6f+3i = 0 =gt 16f + 3i = 5`
`3(11f+2i) - 2(16f+3i) = 12 - 10`
`33f + 6i - 32f - 6i = 2`
`f = 2 =gt c = 4 - 1 = 3`
`12+6+2i = 0 =gt 2i = -18 =gt i = -9`
Hence, evaluating the matrix B under given conditions yields `B = ((3,-4,3),(1,-2,2),(-7,11,-9)).`
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