answer:Suppose, for the sake of contradiction, that every matrix in B commutes with each other. Then, for any matrix C in the vector space, we can write C as a linear combination of the matrices in B: C = sum_{k=1}^{n} c_k B_k where c_k are scalars. Now, for any B_j in B, we have: B_j C = B_j left( sum_{k=1}^{n} c_k B_k right) = sum_{k=1}^{n} c_k B_j B_k Since the matrices in B commute, we can rearrange the order of multiplication: B_j C = sum_{k=1}^{n} c_k B_k B_j = C B_j This implies that every matrix in B commutes with C. But this contradicts our assumption that there exist at least two matrices that do not commute. Therefore, our initial assumption must be false, and there must exist at least two matrices in B that do not commute.

answer:We can use the energy conservation equation to find the force constant: {eq}U_i + K_i = U_f + K_f frac{1}{2} k x_0^2 + frac{1}{2} m v_0^2 = frac{1}{2} k x^2 + frac{1}{2} m v^2 {/eq} Assuming that the maximum distance (amplitude) has zero velocity: {eq}frac{1}{2} k x_0^2 = frac{1}{2} m v_0^2 k = dfrac{m v_0^2}{x_0^2} {/eq} Plugging in the values: {eq}k = dfrac{(0.150text{ kg})(0.450text{ m/s})^2}{(0.080text{ m})^2} boxed{k = 4.73text{ N/m}} {/eq}

answer:When the right string is cut, the rod experiences gravitational force ( F_g = mg ) acting downwards through its center of mass and tension ( T ) from the left string. The system undergoes both linear and rotational motion. We can use Newton's laws and the parallel axes theorem to analyze this situation. a) To find the angular acceleration (( alpha )), we use the torque equation relative to the pivot point: [ tau_{F_g} = F_g cdot frac{l}{2} = I cdot alpha ] where ( I ) is the moment of inertia about the pivot. Using the parallel axes theorem: [ I = I_{CM} + frac{ml^2}{4} ] [ I_{CM} = frac{1}{12}ml^2 ] Combining these, we get: [ I = frac{1}{12}ml^2 + frac{1}{4}ml^2 = frac{1}{3}ml^2 ] Solving for ( alpha ): [ alpha = frac{mg cdot frac{l}{2}}{I} = frac{3g}{2l} ] Now, the linear acceleration of any point on the rod is related to the angular acceleration by: [ a_P = alpha cdot l_P ] 1. For the center of mass (middle of the rod): [ a_{CM} = alpha cdot frac{l}{2} = frac{3}{4}g ] 2. For the right end of the rod: [ a_R = alpha cdot l = frac{3}{2}g ] b) To determine the tension in the left string, we use the linear acceleration of the center of mass in Newton's second law: [ F_g - T = ma_{CM} ] Substituting ( a_{CM} ) and simplifying: [ mg - T = m cdot frac{3}{4}g ] [ T = mg - frac{3}{4}mg ] [ T = frac{1}{4}mg ] Therefore, the linear accelerations are ( a_{CM} = frac{3}{4}g ) for the middle of the rod and ( a_R = frac{3}{2}g ) for the end of the rod. The tension in the left string immediately after the cut is ( T = frac{1}{4}mg ).

answer:The main difference between the two income statements is the treatment of fixed manufacturing overhead. Under variable costing, fixed manufacturing overhead is treated as a period cost and is expensed in the period incurred. Under absorption costing, fixed manufacturing overhead is treated as a product cost and is capitalized as part of inventory. This difference in treatment leads to a higher reported net income under variable costing than under absorption costing. a. Reported income for management: Sales: 5,000 service contracts x 100 = 500,000 Variable cost of goods sold: Beginning inventory: 1,000 service contracts x 90 = 90,000 Direct labor: 4,000 service contracts x 25 = 100,000 Variable overhead: 4,000 service contracts x 5 = 20,000 Total variable cost of goods sold: 210,000 Contribution margin: 500,000 - 210,000 = 290,000 Fixed manufacturing overhead: 240,000 Fixed selling and administrative expenses: 20,000 Total fixed costs: 260,000 Net income: 290,000 - 260,000 = 30,000 b. Reported income for the bank: Sales: 5,000 service contracts x 100 = 500,000 Cost of goods sold: Beginning inventory: 1,000 service contracts x 90 = 90,000 Current period manufacturing cost: Direct labor: 4,000 service contracts x 25 = 100,000 Variable manufacturing overhead: 4,000 service contracts x 5 = 20,000 Fixed manufacturing overhead: 240,000 Total cost of goods sold: 450,000 Gross profit: 500,000 - 450,000 = 50,000 Operating expenses: Selling and administrative expenses: 20,000 + 5,000 service contracts x 5 = 45,000 Net income: 50,000 - 45,000 = 5,000