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Chapter 4: Problem 342

Let the 3 -digit numbers \(\mathrm{A} 28,3 \mathrm{~B} 9\) and \(62 \mathrm{C}\), where \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are integers between 0 and 9, be divisible by a fixed integer \(\mathrm{k}\), then the determinant $$ \left|\begin{array}{lll} \mathrm{A} & 3 & 6 \\ 8 & 9 & \mathrm{C} \\ 2 & \mathrm{~B} & 2 \end{array}\right| $$ is divisible by... (a) \(3 \mathrm{k}\) (b) \(\mathrm{k}^{-3}\) (c) \(\mathrm{k}\) (d) \((\mathrm{k} / 3)\)

Short Answer

Expert verified
The determinant $$ \left|\begin{array}{ccc} A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2 \end{array}\right| $$ is divisible by \(\boldsymbol{k}\).

Step by step solution

01

Write the determinant of the given 3x3 matrix

The given 3x3 matrix is $$ \begin{bmatrix} A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2 \end{bmatrix}. $$ The determinant of this matrix is: $$ \left|\begin{array}{ccc} A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2 \end{array}\right|. $$
02

Expand the determinant

Now we will expand the determinant using the formula, det(A) = a(ei − fh) − b(di − fg) + c(dh − eg). For this particular matrix, we have: $$ \left|\begin{array}{ccc} A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2 \end{array}\right| = A(9 \cdot 2 - C \cdot B) - 3(8 \cdot 2 - C \cdot 2) + 6(8 \cdot B - 9 \cdot 2). $$ Simplify the expression: $$ \begin{aligned} \left|\begin{array}{ccc} A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2 \end{array}\right| &= 18A - ACB - 48 + 6C + 48B - 108 \\ &= -ACB + 18A + 6C + 48B - 156. \end{aligned} $$
03

Analyze the expanded determinant using divisibility rules

We know that 3-digit numbers A28, 3B9, and 62C are divisible by k. Its relationship with the expanded determinant can be drawn by analyzing the divisibility rules. Since the determinant involves A, C, and B, let's look at their divisibility by k as well: - A28 is divisible by k, which implies A is divisible by k. - 3B9 is divisible by k, which means B is divisible by k. - 62C is divisible by k, which suggests C is divisible by k. Now let's analyze the determinant expression, which is -ACB + 18A + 6C + 48B - 156. Based on the mentioned properties and the expression, we can say that the determinant is divisible by k.
04

Match the result with the given options

We have found that the determinant is divisible by k. Comparing this result with the given options, we see that it matches option (c): (c) \(\boldsymbol{k}\) Therefore the determinant $$ \left|\begin{array}{ccc} A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2 \end{array}\right| $$ is divisible by \(\boldsymbol{k}\).

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Most popular questions from this chapter

If \(z\) is a complex number and \(a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\) are all real, then $$ \left|\begin{array}{lll} a_{1} z+b_{1} z & a_{2} z+b_{2} z & a_{3} z+b_{3} z \\ b_{1} z+a_{1} z & b_{2} z+a_{2} z & b_{3} z+a_{3} z \\ b_{1} z+a_{1} & b_{2} z+a_{2} & b_{3} z+a_{3} \end{array}\right|=\ldots $$ (a) \(|\underline{z}|^{2}\) (b) \(\left(a_{1} a_{2} a_{3}+b_{1} b_{2} b_{3}\right)^{2}|z|^{2}\) (c) c (d) 0

If \(\mathrm{D}_{1}=\left|\begin{array}{ccc}\mathrm{x} & \mathrm{a} & \mathrm{a} \\\ \mathrm{a} & \mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{a} & \mathrm{x}\end{array}\right|\) and \(\mathrm{D}_{2}=\left|\begin{array}{ll}\mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{x}\end{array}\right|\), then... (a) \(D_{1}=3\left(\mathrm{D}_{2}\right)^{(3 / 2)}\) (b) \(D_{1}=3 \mathrm{D}_{2}^{2}\) (c) \((\mathrm{d} / \mathrm{dx})\left(\mathrm{D}_{1}\right)=3 \mathrm{D}_{2}^{2}\) (d) \((\mathrm{d} / \mathrm{dx})\left(\mathrm{D}_{1}\right)=3 \mathrm{D}_{2}\)

\(\mathrm{f}(\mathrm{x})=\left|\begin{array}{ccc}\cos \mathrm{x} & 0 & \sin \mathrm{x} \\ 0 & 1 & 0 \\ -\sin \mathrm{x} & 0 & \cos \mathrm{x}\end{array}\right|, \mathrm{g}(\mathrm{y})=\left|\begin{array}{ccc}\cos \mathrm{y} & -\sin \mathrm{y} & 0 \\ \sin \mathrm{y} & \cos \mathrm{y} & 0 \\ 0 & 0 & 1\end{array}\right|\) (i) \(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})=\) (a) \(\mathrm{f}(\mathrm{xy})\) (b) \(\mathrm{f}(\mathrm{x} / \mathrm{y})\) (c) \(\mathrm{f}(\mathrm{x}+\mathrm{y})\) (d) \(\mathrm{f}(\mathrm{x}-\mathrm{y})\) (ii) Which of the following is correct? (a) \([\mathrm{f}(\mathrm{x})]^{-1}=[1 /\\{\mathrm{f}(\mathrm{x})\\}]\) (b) \([\mathrm{f}(\mathrm{x})]^{-1}=-\mathrm{f}(\mathrm{x})\) (c) \([\mathrm{f}(\mathrm{x})]^{-1}=\mathrm{f}(-\mathrm{x})\) (d) \([\mathrm{f}(\mathrm{x})]^{-1}=-\mathrm{f}(-\mathrm{x})\) (iii) \([\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{y})]^{-1}=\) (a) \(\mathrm{f}\left(\mathrm{x}^{-1}\right) \mathrm{g}\left(\mathrm{y}^{-1}\right)\) (b) \(\mathrm{f}\left(\mathrm{y}^{-1}\right) \mathrm{g}\left(\mathrm{x}^{-1}\right)\) (c) \(\mathrm{f}(-\mathrm{x}) \mathrm{g}(-\mathrm{y})\) (d) \(\mathrm{g}(-\mathrm{y}) \mathrm{f}(-\mathrm{x})\)

The correct match of the following columns is given by \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|} { Column I } & \multicolumn{1}{c|} { Column II } \\\ \hline 1. Leibnitz & A. \(\mathrm{e}^{\mathrm{i} \theta}\) \\ 2\. Euler & B. Mathematical logic \\ 3\. Cayley-Hamilton & C. Calculus \\ 4\. George Boole & D. \(\left(\mathrm{e}^{\mathrm{i} \theta}\right)^{\mathrm{n}}=\mathrm{e}^{\mathrm{i}(\mathrm{n} \theta)}\) \\ 5\. De-moivre & E. Theory of Matrices \\ \hline \end{tabular} (a) \(1-\mathrm{D}, 2-\mathrm{A}, 3-\mathrm{E}, 4-\mathrm{B}, 5-\mathrm{A}\) (b) \(1-\mathrm{B}, 2-\mathrm{D}, 3-\mathrm{A}, 4-\mathrm{C}, 5-\mathrm{E}\) (c) \(1-\mathrm{C}, 2-\mathrm{A}, 3-\mathrm{D}, 4-\mathrm{B}, 5-\mathrm{E}\) (d) \(1-\mathrm{C}, 2-\mathrm{A}, 3-\mathrm{E}, 4-\mathrm{B}, 5-\mathrm{D}\)

The matrix $$ \left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right| $$ is singular if \(\ldots \ldots\) (a) \(a-b=0\) (b) \(a+b=0\) (c) \(a+b+c=0\) (d) \(a=0\)
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