I. The given determinant is \(\begin{vmatrix}1&0&0\\0&1&0\\0&0&1\end{vmatrix}\)
By the definition of minors and cofactors, we have:
M11 = minor of a11=\(\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1
M12 = minor of a12=\(\begin{vmatrix}0&0\\0&1\end{vmatrix}\)=0
M13 = minor of a13 =\(\begin{vmatrix}0&1\\0&0\end{vmatrix}\)=0
M21 = minor of a21 =\(\begin{vmatrix}0&0\\0&1\end{vmatrix}\)=0
M22 = minor of a22 =\(\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1
M23 = minor of a23 =\(\begin{vmatrix}1&0\\0&0\end{vmatrix}\)=0
M31 = minor of a31=\(\begin{vmatrix}0&0\\1&0\end{vmatrix}\)=0
M32 = minor of a32 =\(\begin{vmatrix}1&0\\0&0\end{vmatrix}\)=0
M33 = minor of a33 =\(\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1
A11 = cofactor of a11\(= (−1)^{1+1} M_{11} = 1\)
A12 = cofactor of a12 \(= (−1)^{1+2} M_{12} = 0\)
A13 = cofactor of a13 \(= (−1)^{1+3} M_{13} = 0\)
A21 = cofactor of a21 \(= (−1)^{2+1} M_{21} = 0\)
A22 = cofactor of a22 \(= (−1)^{2+2} M_{22} = 1\)
A23 = cofactor of a23 \(= (−1)^{2+3} M_{23} = 0\)
A31 = cofactor of a31 \(= (−1)^{3+1} M_{31} = 0\)
A32 = cofactor of a32 \(= (−1)^{3+2} M_{32} = 0\)
A33 = cofactor of a33 \(= (−1)^{3+3} M_{33} = 1 \)
(ii) The given determinant is \(\begin{vmatrix}1&0&4\\3&5&-1\\0&1&2\end{vmatrix}\)
By definition of minors and cofactors, we have:
M11 = minor of a11\(=\begin{vmatrix}5&-1\\1&2\end{vmatrix}\)=10+1=11
M12 = minor of a12\(=\begin{vmatrix}3&-1\\0&2\end{vmatrix}\)=6-0=6
M13 = minor of a13 \(=\begin{vmatrix}3&5\\0&1\end{vmatrix}\)=3-0=3
M21= minor of a21 \(=\begin{vmatrix}0&4\\1&2\end{vmatrix}\)=0-4=-4
M22 = minor of a22 =\(=\begin{vmatrix}1&4\\0&2\end{vmatrix}\)=2-0=2
M23 = minor of a23 \(=\begin{vmatrix}1&0\\0&1\end{vmatrix}\)=1-0=1
M31 = minor of a31\(=\begin{vmatrix}0&4\\5&-1\end{vmatrix}\)=0-20=-20
M32 = minor of a32 \(=\begin{vmatrix}1&4\\3&-1\end{vmatrix}\)=-1-12=-13
M33 = minor of a33 \(=\begin{vmatrix}1&0\\3&5\end{vmatrix}\)=5-0=5
A11 = cofactor of a11\(= (−1)^{1+1} M_{11} = 11\)
A12 = cofactor of a12 \(= (−1)^{1+2} M_{12} = −6\)
A13 = cofactor of a13 \(= (−1)^{1+3} M_{13} = 3\)
A21 = cofactor of a21 \(= (−1)^{2+1} M_{21} = 4\)
A22 = cofactor of a23 \(= (−1)^{2+2} M_{22} = 2\)
A23 = cofactor of a23 \(= (−1)^{2+3} M_{23} = −1\)
A31 = cofactor of a31 \(= (−1)^{3+1} M_{31} = −20\)
A32 = cofactor of a32 \(= (−1)^{3+2} M_{32} = 13\)
A33 = cofactor of a33 \(= (−1)^{3+3} M_{33} = 5\)
Let I be the identity matrix of order 3 × 3 and for the matrix $ A = \begin{pmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{pmatrix} $, $ |A| = -1 $. Let B be the inverse of the matrix $ \text{adj}(A \cdot \text{adj}(A^2)) $. Then $ |(\lambda B + I)| $ is equal to _______
Let $A = \{ z \in \mathbb{C} : |z - 2 - i| = 3 \}$, $B = \{ z \in \mathbb{C} : \text{Re}(z - iz) = 2 \}$, and $S = A \cap B$. Then $\sum_{z \in S} |z|^2$ is equal to
If $ y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\27 & 28 & 27 \\1 & 1 & 1 \end{vmatrix} $, $ x \in \mathbb{R} $, then $ \frac{d^2y}{dx^2} + y $ is equal to
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is: