Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by n×n matrices A and B. a)(a+b)2=a2+2ab+b2 b)(a+b)(a−b)=a2−b2

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Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by n×n matrices A and B. a)(a+b)2=a2+2ab+b2  b)(a+b)(a−b)=a2−b2

wornoutwomanC · Accepted Answer

(a)&amp;nbsp;(A+B)2=(A+B)(A+B)(A+B)2=A(A+B)+B(A+B)(A+B)2=AA+AB+BA+BB(A+B)2=A2+AB+BA+B2As&amp;nbsp;n×n&amp;nbsp;matrices are not always commutative, the formula&amp;nbsp;(A+B)2=A2+2AB+B2&amp;nbsp;will not always work in general for all matrices.(b)&amp;nbsp;(A+B)(A−B)=A(A+B)−B(A+B)(A+B)(A−B)=AA+AB−BA−BB(A+B)(A−B)=A2+AB−BA−B2As&amp;nbsp;n×n&amp;nbsp;matrices are not always commutative, the formula&amp;nbsp;(A+B)(A−B)=A2−B2&amp;nbsp;will not always work in general for all matrices.

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Vasquez · Accepted Answer

Step 1:We need to consider the properties and operations of matrices.a) (A+B)2=A2+2AB+B2This rule does not hold for matrices in general. When expanding (A+B)2, we get:(A+B)2=(A+B)(A+B)=A(A+B)+B(A+B)Expanding further, we have:A(A+B)+B(A+B)=A2+AB+BA+B2In general, matrix multiplication is not commutative, so AB and BA are not necessarily equal. Therefore, we cannot simplify the expression to A2+2AB+B2, making the rule invalid for matrices.Step 2:b) (A+B)(A−B)=A2−B2This rule also does not hold for matrices in general. Expanding (A+B)(A−B), we have:(A+B)(A−B)=A(A−B)+B(A−B)Expanding further, we get:A(A−B)+B(A−B)=A2−AB+BA−B2Similarly to the previous case, matrix multiplication is not commutative, so AB and BA are not necessarily equal. Therefore, we cannot simplify the expression to A2−AB+BA−B2, making the rule invalid for matrices.In conclusion, these algebraic rules that hold true for real numbers a and b do not hold in general when matrices A and B are used instead.

RizerMix · Accepted Answer

To explain why the given algebraic rules do not work in general when the real numbers &#039;a&#039; and &#039;b&#039; are replaced by n×n matrices &#039;A&#039; and &#039;B,&#039; we will analyze each rule separately.a) (A+B)2≠A2+2AB+B2Expanding the left-hand side of the equation, we have:(A+B)2=(A+B)(A+B)Using the distributive property, we can expand this expression as follows:(A+B)(A+B)=A(A+B)+B(A+B)Applying the distributive property again, we get:A(A+B)+B(A+B)=A2+AB+BA+B2Notice that in the above expression, we have the terms AB and BA, which represent matrix multiplication between A and B in different orders. In general, matrix multiplication does not commute, meaning that AB and BA can yield different results. Therefore, we cannot simplify A2+AB+BA+B2 to A2+2AB+B2, as we would in the case of real numbers.b) (A+B)(A−B)≠A2−B2Expanding the left-hand side of the equation, we have:(A+B)(A−B)Using the distributive property, we can expand this expression as follows:(A+B)(A−B)=A(A−B)+B(A−B)Applying the distributive property again, we get:A(A−B)+B(A−B)=A2−AB+BA−B2Similar to the previous case, we have the terms AB and BA, which represent matrix multiplication between A and B in different orders. Since matrix multiplication does not generally commute, we cannot simplify A2−AB+BA−B2 to A2−B2.In conclusion, the algebraic rules (A+B)2=A2+2AB+B2 and (A+B)(A−B)=A2−B2 do not hold in general when real numbers &#039;a&#039; and &#039;b&#039;&#039; are replaced by n×n matrices &#039;A&#039; and &#039;B&#039; due to the non-commutativity of matrix multiplication.

Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by n times n matrices A and B. a) (a+b)^2=a^2+2ab+b^2 b) (a+b)(a-b)=a^2-b^2

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