Advertisements
Advertisements
Question
The sum of the products of elements of any row with the co-factors of corresponding elements is equal to ______.
Advertisements
Solution
The sum of the products of elements of any row with the co-factors of corresponding elements is equal to the value of the determinant of the given matrix.
Explanation:
Let Δ = `|("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)|`
Expanding along R1
`"a"_11 |("a"_22, "a"_23),("a"_32, "a"_33)| - "a"_12 |("a"_21, "a"_23),("a"_31, "a"_33)| + "a"_13 |("a"_21,"a"_22),("a"_31, "a"_32)|`
⇒ `"a"_11"M"_11 + "a"_12"M"_12 + "a"_13"M"_13` ....(Where M11, M12 and M13 are the minors of the corresponding elements)
APPEARS IN
RELATED QUESTIONS
Write Minors and Cofactors of the elements of the following determinant:
`|(a,c),(b,d)|`
Write Minors and Cofactors of the elements of the following determinant:
`|(1,0,4),(3,5,-1),(0,1,2)|`
Using Cofactors of elements of second row, evaluate Δ = `|(5,3,8),(2,0,1),(1,2, 3)|`.
Using Cofactors of elements of third column, evaluate Δ = `|(1,x,yz),(1,y,zx),(1,z,xy)|`.
if A = `((2,3,10),(4,-6,5),(6,9,-20))`, Find `A^(-1)`. Using `A^(-1)` Solve the system of equation `2/x + 3/y +10/z = 2`; `4/x - 6/y + 5/z = 5`; `6/x + 9/y - 20/z = -4`
Using matrices, solve the following system of equations :
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}5 & 20 \\ 0 & - 1\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}- 1 & 4 \\ 2 & 3\end{bmatrix}\]
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{bmatrix}\]
If \[A = \begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}\] and Cij is cofactor of aij in A, then value of |A| is given
Write the adjoint of the matrix \[A = \begin{bmatrix}- 3 & 4 \\ 7 & - 2\end{bmatrix} .\]
If Cij is the cofactor of the element aij of the matrix \[A = \begin{bmatrix}2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7\end{bmatrix}\], then write the value of a32C32.
Write \[A^{- 1}\text{ for }A = \begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
If \[A = \begin{bmatrix}5 & 6 & - 3 \\ - 4 & 3 & 2 \\ - 4 & - 7 & 3\end{bmatrix}\] , then write the cofactor of the element a21 of its 2nd row.
If A = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]`, find A–1. Using A–1, solve the system of linear equations x – 2y = 10, 2x – y – z = 8, –2y + z = 7.
Using matrix method, solve the system of equations
3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2.
If A `= [(0,1,1),(1,0,1),(1,1,0)] "then" ("A"^2 - 3"I")/2 =` ____________.
Evaluate the determinant `Delta = abs (("log"_3 512, "log"_4 3),("log"_3 8, "log"_4 9))`
`abs(("cos" 15°, "sin" 15°),("sin" 75°, "cos" 75°))`
Find the minor of 6 and cofactor of 4 respectively in the determinant `Delta = abs ((1,2,3),(4,5,6),(7,8,9))`
For a square matrix \(A=[a_{ij}]\) of order \(n\), what is the minor \(M_{ij}\) of \(a_{ij}\)?
What is the first step to find the minor \(M_{ij}\)?
What calculation gives the resulting value \(M_{ij}\) after the required row and column have been deleted?
Which multiplication completes the calculation of a cofactor?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{11}\)?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{12}\)?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{21}\)?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(M_{22}\)?
For \[\begin{vmatrix}1&-2\\4&3\end{vmatrix}\], what is \(A_{11}\)?
Which expression gives determinant expansion along row \(i\)?
Which expression gives determinant expansion along column \(j\)?
