If A = [[3, 0], [5, 1]] and B = [[-4, 2], [1, 0]], find A² - 2AB + B²

Given:

A = [[3,0],[5,1]]

B = [[−4,2],[1,0]]

Required: A² − 2AB + B²

First calculate A²:

A² = [[3,0],[5,1]] [[3,0],[5,1]]

= [[3×3 + 0×5, 3×0 + 0×1], [5×3 + 1×5, 5×0 + 1×1]]

= [[9,0],[20,1]]

Now calculate AB:

AB = [[3,0],[5,1]] [[−4,2],[1,0]]

= [[3(−4)+0(1), 3(2)+0(0)], [5(−4)+1(1), 5(2)+1(0)]]

= [[−12,6],[−19,10]]

Therefore:

−2AB = [[24,−12],[38,−20]]

Now calculate B²:

B² = [[−4,2],[1,0]] [[−4,2],[1,0]]

= [[(−4)(−4)+2(1), (−4)(2)+2(0)], [1(−4)+0(1), 1(2)+0(0)]]

= [[18,−8],[−4,2]]

Hence:

A² − 2AB + B²

= [[9,0],[20,1]] + [[24,−12],[38,−20]] + [[18,−8],[−4,2]]

= [[51,−20],[54,−17]]

Answer:

[[51,−20],[54,−17]].

Calculate A², AB and B² separately by row-by-column multiplication. Then form A² − 2AB + B² by adding corresponding entries.