Comprehension

A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

Question: 1

Write the equations representing the various dimensions and express them as the matrix equation \(AX = B\).

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Always align variable names columns (\(x\), then \(y\), then \(z\)) before reading off the coefficients. Mixing up the order of variables while reading word problems is the most frequent cause of incorrect matrix setups.

Updated On: Jul 8, 2026

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Question: 2

Find if \(A^{-1}\) exists. Justify your answer.

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When expanding determinants, pay close attention to the alternating signs (\(+, -, +\)) assigned to the row positions. Forgetting the minus sign for the middle term is the single most common math arithmetic pitfall.

Updated On: Jul 8, 2026

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Question: 3

Find \(A^{-1}\).

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To double check your inverse accuracy, multiply a single row of the original matrix by a column of your computed adjugate matrix. It should always equal the determinant value (\(-8\)). For instance, row 1: \(1(20) + 1(-17) + (-1)(11) = 20 - 17 - 11 = -8\). Perfect!

Updated On: Jul 8, 2026

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Question: 4

Find \(A^2 + 7I\).

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Adding a scalar identity matrix like \(7I\) only updates the values sitting directly on the main diagonal axis of your target matrix. The rest of the off-diagonal cells remain completely untouched, saving significant execution time!

Updated On: Jul 8, 2026

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