This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. 2. Commutative Law of Subtraction The commutative law of addition states that A+B = B+A. Matrix subtraction A-B is defined as A+(-1)B. If we assume A-B = B-A, then A+(-1)B = B+(-1)A. This statement is generally not true for matrices. For example, if A = 1 & 0 \\ 0 & 0 and B = 0 & 1 \\ 0 & 0 : A-B = 1 & -1 \\ 0 & 0 B-A = -1 & 1 \\ 0 & 0 Clearly, A-B ≠ B-A. The commutative law of addition does not apply directly to subtraction in this way. To correct the statement, we can say that matrix subtraction is anti-commutative, meaning: A-B = -(B-A) Let's verify this: -(B-A) = -(B+(-1)A) = -B - (-1)A = -B + A = A-B This shows that A-B = -(B-A) is the correct relationship. Does the commutative law of addition permit us to state that A-B = B-A? ❌ No. How would you correct the statement? The correct statement is A-B = -(B-A). 3. Test the associative law of multiplication with the following matrices: Given matrices: A = 5 & 3 \\ 0 & 5 , B = -8 & 0 & 7 \\ 1 & 3 & 2 , C = 1 \\ 0 \\ 7 The associative law of multiplication states that (AB)C = A(BC). Step 1: Calculate AB. AB = 5 & 3 \\ 0 & 5 -8 & 0 & 7 \\ 1 & 3 & 2 AB = (5)(-8)+(3)(1) & (5)(0)+(3)(3) & (5)(7)+(3)(2) \\ (0)(-8)+(5)(1) & (0)(0)+(5)(3) & (0)(7)+(5)(2) AB = -40+3 & 0+9 & 35+6 \\ 0+5 & 0+15 & 0+10 AB = -37 & 9 & 41 \\ 5 & 15 & 10 Step 2: Calculate (AB)C. (AB)C = -37 & 9 & 41 \\ 5 & 15 & 10 1 \\ 0 \\ 7 (AB)C = (-37)(1)+(9)(0)+(41)(7) \\ (5)(1)+(15)(0)+(10)(7) (AB)C = -37+0+287 \\ 5+0+70 (AB)C = 250 \\ 75 Step 3: Calculate BC. BC = -8 & 0 & 7 \\ 1 & 3 & 2 1 \\ 0 \\ 7 BC = (-8)(1)+(0)(0)+(7)(7) \\ (1)(1)+(3)(0)+(2)(7) BC = -8+0+49 \\ 1+0+14 BC = 41 \\ 15 Step 4: Calculate A(BC). A(BC) = 5 & 3 \\ 0 & 5 41 \\ 15 A(BC) = (5)(41)+(3)(15) \\ (0)(41)+(5)(15) A(BC) = 205+45 \\ 0+75 A(BC) = 250 \\ 75 Step 5: Compare the results. Since (AB)C = 250 \\ 75 and A(BC) = 250 \\ 75 , we have (AB)C = A(BC). The associative law of multiplication is verified for these matrices. Verified