0
How evaluate the following determinant by row operations 3 2 6 1 1 2 2 2 5?
Wiki User
∙ 15y ago
determinant sign | |
3 2 6 + - +
1 1 2 - + -
2 2 5 + - + (don't forget the signs)
Work with the second row:
= (-1)|2 6, 2 5| + 1|3 6, 2 5| - 2|3 2, 2 2| (draw a vertical line, and a horizontal one at 1, 1, and 2 in the second row)
= -1[2 x 5) - (6 x 2)] + [(3 x 5) - (6 x 2)] - 2[(3 x 2) - (2 x 2)]
= (-1)(10 - 12) + (15 - 12) - (2)(6 - 4)
= (-1)(-2) + 3 - (2)(2)
= 2 + 3 - 4
= 1
Thus the value of the determinant is equal to 1.
Wiki User
∙ 15y ago
Add your answer:
Earn +20 pts
What is a cofactor of a determinant?
The cofactor is the signed minor of a determinant, used to evaluate the determinant. You take the minor of the element - call that element aij - and if i + j is even, the cofactor is the minor - otherwise, it's the opposite of the minor. Thus, take the matrix, remove the row and column the element is in, and if the sum of the row number and column number is even, then there's your cofactor; otherwise, it's the additive inverse. For example, the cofactor of a34 is the determinant of the same matrix with the 3rd row and 4th column removed, and then you take the opposite (additive inverse or negative), because 3 + 4 = 7 is odd.
What is the minor of determinant?
The minor is the determinant of the matrix constructed by removing the row and column of a particular element. Thus, the minor of a34 is the determinant of the matrix which has all the same rows and columns, except for the 3rd row and 4th column.
What is the determinant of a two-by-two matrix?
To find that, you multiply the first element of the first row by the second element of the second row. You also multiply the first element of the second row with the second element of the first row. Then you subtract the products not add them.
What is the physical meaning of determinant of a matrix?
for a 3x3 matrix, it can be interpreted as the volume of the hexahedron formed by three vectors (each row of the matrix as one vector).
What is the formula for determinant of a 3x3 matrix?
To answer this question, let me establish an example 3 x 3 matrix named "A": A= [a b c] [d e f] [g h i] The formula I will give you, called co-factor expansion, works for any size square matrix, so you could use it to find the determinant of a 2 x 2, 3 x 3, all the way up to an n x n matrix. To find the determinant, pick any row or column in the matrix. It will make your work much easier if you choose a row or column that has many zeroes in it. A general notation that is often used to find the determinant of a matrix is to use straight bars in place of the brackets surrounding the matrix contents. So, if I was to say mathematically that I was finding the determinant of the above example matrix, I could write it as: det(A)= |a b c| |d e f| |g h i| This notation will be used in the formula, so it is important to know this. For the sake of an arbitrary example, let us suppose I chose Row 1 of the matrix as my chosen row. To find the determinant of this matrix, I will perform the following calculation: (-1)2(a)|e f| + (-1)3(b)|d f| + (-1)4(c)|d e| |h i| |g i| |g h| This is the specific application of this general formula to the example matrix: (-1)i+j(aij)det(A1) In this formula, i and j are the row and column addresses, respectively, of a given matrix element. So, like in our specific application, when Row 1 was chosen as our subject row, the first term was (-1)1+1(A11)det(A1). The element "a" is in the first row, first column of the matrix, mean i=1 and j=1, therefore the superscript of (-1) is 1+1=2. A11 is simply the value held in the address i=1, j=1 of the matrix A. For this application, A11 was "a". det(A1) is the determinant of the submatrix A1. This submatrix has no formal nomenclature, I simply call it this for ease of explanation. A1 is the matrix created by "crossing out" the row and column that belong to the matrix element A11. In this application, that means it is the submatrix that is left after crossing out a, b, c, d, and g, which is simply the 2 x 2 matrix e,f;h,i. Performing this same process for the remainder of the matrix elements in Row 1 will yield the determinant of the matrix. So, the "generalized" form of the specific application above is: (-1)1+1(A11)det(A1) + (-1)1+2(A12)det(A2) + (-1)1+3(A13)det(A3) where A1 is the submatrix created by crossing out Row 1 and Column 1, A2 is the submatrix created by crossing out Row 1 and Column 2, and A3 is the submatrix created by crossing out Row 1 and Column 3. A final note is how to calculate the determinants of the submatrices. For a 3 x 3 matrix, its submatrices are all 2 x 2. For 2 x 2 matrices, a simple formula exists that makes this easy: |a b| = (ad) - (bc) |c d| For higher-dimension matrices, the submatrices also become larger, making the computation much more intensive.
Does doing row operations of a matrix change its determinant?
No.
Does doing row operations on a matrix change its determinant?
No.
What is a cofactor of a determinant?
The cofactor is the signed minor of a determinant, used to evaluate the determinant. You take the minor of the element - call that element aij - and if i + j is even, the cofactor is the minor - otherwise, it's the opposite of the minor. Thus, take the matrix, remove the row and column the element is in, and if the sum of the row number and column number is even, then there's your cofactor; otherwise, it's the additive inverse. For example, the cofactor of a34 is the determinant of the same matrix with the 3rd row and 4th column removed, and then you take the opposite (additive inverse or negative), because 3 + 4 = 7 is odd.
What is the minor of determinant?
The minor is the determinant of the matrix constructed by removing the row and column of a particular element. Thus, the minor of a34 is the determinant of the matrix which has all the same rows and columns, except for the 3rd row and 4th column.
What is the determinant of a two-by-two matrix?
To find that, you multiply the first element of the first row by the second element of the second row. You also multiply the first element of the second row with the second element of the first row. Then you subtract the products not add them.
What is the meaning of determinant of a matrix?
for a 3x3 matrix, it can be interpreted as the volume of the hexahedron formed by three vectors (each row of the matrix as one vector).
What is gauss eliminating method?
Consider a system of linear equations . Let be its coefficient matrix. elementary row operation.(i) R(i, j): Interchange of the ith and jth row.(ii) R(ci): Multiplying the ith row by a non-zero scalar c.(iii) R(i, cj): Adding c times the jth row to the ith row.It is clear that performing elementary row operations on the matrix (or on the equations themselves) does not affect the solutions. Two matrices and are said to be row equivalent if and only if one of them can be obtained from the other by performing a sequence of elementary row operations. A matrix is said to be in row echelon form the following conditions are satisfied:(i) The number of first consecutive zerosincreases down the rows.(ii) The first non-zero element in each row is 1.The process of performing a sequence of elementary row operations on a system of equations so that the coefficient matrix reduces to row echelon form is called Gauss elimination. When a system of linear equations is transformed using elementary row operations so the coefficient matrix is in row echelon form, the solution is easily obtained by back substitution.
What is the physical meaning of determinant of a matrix?
for a 3x3 matrix, it can be interpreted as the volume of the hexahedron formed by three vectors (each row of the matrix as one vector).
If the determinant of a 5x5 matrix delta is the detdelta 5 and the matrix feta is obtained from delta by adding 9 times the third row to the second then detfeta is?
It is the same.
How do you get inverse of a matrix?
There are two main methods: both depend on a reasonable understanding of matrices and determinants.One way is to append an identity matrix to the right of the original matrix. So, given an n x n matrix A, create an n x 2n matrix A|I.Using only row operations, convert A into an identity matrix. Then the right hand side of A|I will be A-1Alternatively, calculate the determinant |A|Also calculate the determinants of all n-1 x n-1 sub-matrices where sijdenotes the determinant of the submatrix created when row i and column j are deleted.Let bij= sij*(-1)i+j/|A|.Then the matrix B = {bij} = A-1.
What's the determinant of a matrix?
A determinant is defined for square matrices only.To find the determinant of the matrix you need to:find all n-tuples of elements of the matrix such that each row and each column of the matrix is represented.calculate the product of the elements.calculate the sign for that term. To see how this is done, see below.calculate the sum of the signed products: that is the determinant.To calculate the sign for the product of the n-tuple, arrange the elements in row order. Swap the elements, two at a time, to get them in column order. If the number of swaps required is even then the product is assigned a positive sign, and if odd then a negative sign.
What is the determinant of a matrix?
A determinant is defined for square matrices only.To find the determinant of the matrix you need to:find all n-tuples of elements of the matrix such that each row and each column of the matrix is represented.calculate the product of the elements.calculate the sign for that term. To see how this is done, see below.calculate the sum of the signed products: that is the determinant.To calculate the sign for the product of the n-tuple, arrange the elements in row order. Swap the elements, two at a time, to get them in column order. If the number of swaps required is even then the product is assigned a positive sign, and if odd then a negative sign.
