Question
Find minors and cofactors of the elements a11, a21 in the determinant $\Delta=\left|\begin{array}{lll} {a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}} \end{array}\right|$
✓
Answer
By definition of minors and cofactors, we have
Minor of a11 = M11 = $\left|\begin{array}{ll} {a_{22}} & {a_{23}} \\ {a_{32}} & {a_{33}} \end{array}\right|$ = a22 a33 – a23 a32
Cofactor of a11 = A11 = (–1)1+1 M11 = a22 a33 – a23 a32
Minor of a21 = M21 = $\left|\begin{array}{ll} {a_{12}} & {a_{13}} \\ {a_{32}} & {a_{33}} \end{array}\right|$ = a12 a33 – a13 a32
Cofactor of a21 = A21 = (–1)2+1 M21 = (–1) (a12 a33 – a13 a32) = – a12 a33 + a13 a32
Minor of a11 = M11 = $\left|\begin{array}{ll} {a_{22}} & {a_{23}} \\ {a_{32}} & {a_{33}} \end{array}\right|$ = a22 a33 – a23 a32
Cofactor of a11 = A11 = (–1)1+1 M11 = a22 a33 – a23 a32
Minor of a21 = M21 = $\left|\begin{array}{ll} {a_{12}} & {a_{13}} \\ {a_{32}} & {a_{33}} \end{array}\right|$ = a12 a33 – a13 a32
Cofactor of a21 = A21 = (–1)2+1 M21 = (–1) (a12 a33 – a13 a32) = – a12 a33 + a13 a32
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2).
Write the value of $\cos^{-1}(\cos1540^\circ).$
→Evaluate the following definite integrals:
$\int_{0}^\limits{\infty}\text{e}^{-\text{x}}\text{ dx}$
→$\int_{0}^\limits{\infty}\text{e}^{-\text{x}}\text{ dx}$
If $\text{A}=\begin{bmatrix}\cos\text{x}&-\sin\text{x}\\\sin\text{x}&\cos\text{x}\end{bmatrix},$ find AAT.
→If I is the identity matrix and A is a square matrix such that A2 = A, then what is the value of (I + A)2 = 3A?
Find the vector equation of a plane which is at a distance of 3 units from the origin and has $\hat{\text{k}}$ as the unit vector normal to it.
→Write the identity element for the binary operation * defined on the set R of all real numbers by the rule $\text{a}\times\text{b}=\frac{3\text{ab}}{7}\ \forall\text{ a, b}\in\text{R}$.
→Write the points of non-differentiability of f(x) = |log |x||.
For a 2×2 matrix A = [aij] whose elements are given by $\text{a}_{\text{ij}}=\frac{\text{i}}{\text{j}}$
, write the value of a12.
→, write the value of a12.Evalute the following integrals:
$\int\sqrt{\frac{1+\cos2\text{x}}{1-\cos2\text{x}}}\text{dx}$
→$\int\sqrt{\frac{1+\cos2\text{x}}{1-\cos2\text{x}}}\text{dx}$