Let P be a matrix of order 3 × 3 such that all entries in P are from set {-1,0,1} . Then, maximum p

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3 and 4 .Determinants and Matrices

normal

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1,0,1\}$. Then, the maximum possible value of the determinant of $P$ is. . . . . . .

$7$

$6$

$5$

$4$

(IIT-2018)

Solution

$\Delta=\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=\left(a_1 b_2 c_3+a_3 c_2 b_1+a_2 c_1 b_3\right)-\left(a_1 b_3 c_2+a_2 c_3 b_1+a_3 c_1 b_2\right)$

$\left\{ a _{ i }, b _{ i }, c _{ i }\right\} \equiv\{-1,0,1\} i =1,2,3$

Maximum value can be $6$, but that's not possible.

(For $\Delta=6,3$ entries should be $1$ and $6$ entries should be -$1$ in first bracket and $3$ entries should be -$1$ and $6$ entries should be $1$ in second bracket. That's a contradiction!)

$\Delta \neq 5$. (Even if one element is $0$ )

$\therefore \Delta_{\max }=4$ which is possible.

$\operatorname{Ex}:\left[\begin{array}{ccc}1 & -1 & 0 \\ 1 & 1 & 1 \\ -1 & -1 & 1\end{array}\right]$

Standard 12

Mathematics

The value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&{a\, + \,b}&{a\, + \,2b}\\{a\, + \,2b}&a&{a\, + \,b}\\{a\, + \,b}&{a\, + \,2b}&a\end{array}\,} \right|$ is

medium

If $f(x) = \left| {\begin{array}{*{20}{c}}1&x&{x + 1}\\{2x}&{x(x – 1)}&{(x + 1)x}\\{3x(x – 1)}&{x(x – 1)(x – 2)}&{(x + 1)x(x – 1)}\end{array}} \right|$ then $f(100)$ is equal to

hard

(IIT-1999)

$\left| {\begin{array}{*{20}{c}}{1 + {{\sin }^2}\theta }&{{{\sin }^2}\theta }&{{{\sin }^2}\theta }\\{{{\cos }^2}\theta }&{1 + {{\cos }^2}\theta }&{{{\cos }^2}\theta }\\{4\sin 4\theta }&{4\sin 4\theta }&{1 + 4\sin 4\theta }\end{array}} \right| = 0$ then $\sin \,4\theta $ equal to

hard

If $a, b, c,$ are non zero complex numbers satisfying $a^2 + b^2 + c^2 = 0$ and $\left| {\begin{array}{*{20}{c}}
{{b^2} + {c^2}}&{ab}&{ac}\\
{ab}&{{c^2} + {a^2}}&{bc}\\
{ac}&{bc}&{{a^2} + {b^2}}
\end{array}} \right| = k{a^2}{b^2}{c^2},$ then $k$ is equal to

hard

(AIEEE-2012)

Which of the following matrices can $NOT$ be obtained from the matrix $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$ by a single elementary row operation?

easy

(JEE MAIN-2022)

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1,0,1\}$. Then, the maximum possible value of the determinant of $P$ is. . . . . . .

Solution

Similar Questions

The value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&{a\, + \,b}&{a\, + \,2b}\\{a\, + \,2b}&a&{a\, + \,b}\\{a\, + \,b}&{a\, + \,2b}&a\end{array}\,} \right|$ is

If $f(x) = \left| {\begin{array}{*{20}{c}}1&x&{x + 1}\\{2x}&{x(x – 1)}&{(x + 1)x}\\{3x(x – 1)}&{x(x – 1)(x – 2)}&{(x + 1)x(x – 1)}\end{array}} \right|$ then $f(100)$ is equal to

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If $a, b, c,$ are non zero complex numbers satisfying $a^2 + b^2 + c^2 = 0$ and $\left| {\begin{array}{*{20}{c}}
{{b^2} + {c^2}}&{ab}&{ac}\\
{ab}&{{c^2} + {a^2}}&{bc}\\
{ac}&{bc}&{{a^2} + {b^2}}
\end{array}} \right| = k{a^2}{b^2}{c^2},$ then $k$ is equal to