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One more question. The total no.of onto function from the set {a,b,c,d,e,f} to the set {1,2,3} is????? 1 The total no.of onto function from the set {a,b,c,d,e,f} to the set {1,2,3} is 2 The adjoint of 3 by 3 matrix A is then the absolute value of the determinant A is 3 for any 3 by 3 matrix A If A(adj A)= then the value of |A| is 4 If A is a square matrix such that aij=(i+j)(i-j) then for A to be non-singular the order of matrix can be

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@Kajal

\mathbf{{\color{DarkBlue} |Adj A|=|A|^{n-2}}}

\\|AdjA|=\begin{bmatrix} 3 &-2 &7 \\ 2&-6 &-4 \\ 1&3 &2 \end{bmatrix}=100\\|A|^2=100\\|A|=\pm10\\Absolute\:value\:is\:10

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himanshu.meshram

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@KAJAL
\\A(AdjA)=|A|I_n\\given\:\:A(AdjA)=\begin{bmatrix} 7 & 0 &0 \\ 0& 7 &0 \\ 0& 0& 7 \end{bmatrix}\\ now, by \;formula\;|A|I_n=\begin{bmatrix} 7 & 0 &0 \\ 0& 7 &0 \\ 0& 0& 7 \end{bmatrix}\\|A|I_n=7\begin{bmatrix} 1 & 0 &0 \\ 0& 1 &0 \\ 0& 0& 1 \end{bmatrix}\\so \; on\: compairing\: we \: get \: |A|=7.???????

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Pankaj

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@Kajal

\\a_{ij}=(i+j)(i-j)=i^2-j^2\\if\:order\:matrix\:is\:odd\:then\:matrix \:A\:is \:skew\:symmetric \:matrix.\\e.g.\:take \:3\times3\:matrics\\given\:a_{ij}=i^2-j^2\\A_{3\times3}=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix}\\\Rightarrow \begin{pmatrix}1^2-1^2&1^2-2^2&1^2-3^2\\ 2^2-1^2&2^2-2^2&2^2-3^2\\ 3^2-1^2&3^2-2^2&3^2-3^2\end{pmatrix}=\begin{pmatrix}0&-3&-8\\ 3&0&-5\\ 8&5&0\end{pmatrix}

\mathbf{{\color{DarkBlue} The \:determinant\: of\: an \:n\times n\: skew-symmetric\: matrix \:is\: zero\: \:if\:\: n \:is\: odd.}}

So, for matrix has to be non-singular order should be 2.

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himanshu.meshram

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@Kajal

{\color{DarkGreen} The \:total\: no.\:of \:onto \:function \:from \:the \:set \:({a,b,c,d,e,f})\: to\: the\: set\:( {1,2,3})\: is?}

\\E=\begin{Bmatrix} a,b,c,d,e,f \end{Bmatrix}\\F=\begin{Bmatrix} 1,2,3 \end{Bmatrix}\\n(E)=6\\n(F)=3\\m>n

\mathbf{{\color{DarkBlue} no\:of\:onto\:function\:=\sum _{r=1}^n\left(-1\right)^{n-r}\cdot ^nC_rr^m}}

\Rightarrow \sum _{r=1}^3\left(-1\right)^{3-r}\cdot ^3C_rr^6=\left(-1\right)^{3-1}\cdot ^3C_11^6+\left(-1\right)^{3-2}\cdot ^3C_22^6+\left(-1\right)^{3-3}\cdot ^3C_33^6

\Rightarrow 540

Posted by

himanshu.meshram

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