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relationship between determinant and adjoint
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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 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.
What is an eigenvector?
Given some matrix A, an eigenvector of A is a vector that, when acted on by A, will result in a scalar multiple of itself, i.e. Ax=[lambda]x, where lambda is a real scalar multiple, called an eigenvalue, and x is the eigenvector described.To find x you will normally have to find lambda first, which means solving the "characteristic equation": det(A-[lambda]I)=0, where I is the identity matrix.The derivation of the "characteristic equation" is as follows:Rearrange the equation Ax=[lambda]x -> Ax-[lambda]x=0 -> (A-[lambda]I)x=0 and then use the property from linear algebra that says if (A-[lambda]x) has an inverse, then x=0. Since this is trivial, we must instead prove that (A-[lambda]x) does not have an inverse. Because the inverse of a matrix is equal to its transpose divided by its determinant, and because you can't divide by 0, a 0 valued determinant means that the inverse can't exist. This is why we must solve det(A-[lambda]I)=0 for lambda.Once we have found lambda, we can put it in the equation Ax=[lambda]x, and it's then just a simple matter of solving the resulting linear equations.
Find the value of k for which the equation 2x2 - parenthesis k - 1 parenthesis x plus 80 equals 0 will have real and equal roots?
A quadratic of the form ax² + bx + c = 0 has two equal (real) roots when the determinant of b²-4ac = 0. Thus, for: 2x² - (k - 1)x + 80 = 0 b² - 4ac = (-(k - 1))² - 4×2×80 = 0 → (k - 1)² = 640 → k - 1 = ±√640 → k = 1 ± √640 → k ≈ 26.298 or -24.298
How do you find missing frequency if median and mode are given?
How do you find missed frequency if median and mode are given
How to find x in a 2×2 matrix with a given determinant.?
For a matrix A, A is read as determinant of A and not, as modulus of A. ... sum of two or more elements, then the given determinant can be expressed as the sum
What is meant by the determinant of a math equation?
A single math equation does not have a determinant. A system of equations (3x3 , 4x4, etc.) will have a determinant. You can find a determinant of a system by converting the system into a corresponding matrix and finding its determinant.
Find the determinant of 1 a AA AA 1 a a AA 1?
Assuming that the terms, a and AA, are commutative, It is 1 + a^3 + (AA)^3 - 3aAA
How do you find the determinant of a square matrix?
233
Is there an easy way by which I mean a way that requires less than half an hour and no higher math than Algebra 2 to find the determinant of a 5 by 5 matrix?
There is no easy way to find the determinant; it's long and tedious. There are computer programs available (like MATLAB) that will find the determinant. You'll find there probably won't be a large matrix in an exam if you're required to find the determinant.
How can you solve if the determinant of 3 by 3 matrix is 2?
It isn't clear what you want to solve for. If you want to find the matrix, there is not a unique solution - there are infinitely many matrices with the same determinant.
How do you find the variable in this matrix 3 -2 5 a ...the determinant is 7?
If it a 2x2 matrix, the determinant is 3*a - (-2)*5 = 3a + 10 = 7 So 3a = -3 so a = -1
What determines the nature of the roots of a quadratic equation?
The determinant.The determinant is the part under the square root of the quadratic equation and is:b2-4ac where your quadratic is of the form: ax2+bx+cIf the determinant is less than zero then you have 'no real solutions' (as the square root of a negative number is imaginary.)If the determinant is = 0, then you have one real solution (because you can discount the square root of the quadratic equation)If the determinant is greater than zero you have two real solutions as you have (-b PLUS OR MINUS the square root of the determinant) all over 2aTo find the solutions where they exist you'll need to solve the quadratic formula or use another method.
What is the solution of this question By wronskian determinant method please find whether the solution exist or not where.... y and 8321cos and sup2x y and 83221 plus cos2x?
What is the solution of this question By wronskian determinant method please find whether the solution exist or not where.... y1 = (squre of) cos x, y2 = 1 + cos 2 x.
Find the rank of the determinant of 1 2 3 4 5 6 its a 2 x 3 matrics?
This cannot be answered. This does not make any sense.
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.
