# Investing a triangular matrix determinant

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How many of each type of tomato do you have? What percentage does each vegetable have in the market share? Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges.

For each fruit, find the percentage of total fruit sold. Three bands performed at a concert venue. If the first band had 40 more audience members than the second band, how many tickets were sold for each band? A movie theatre sold tickets to three movies. How many tickets for each movie were sold? This year, the same age groups made up Determine the prison population percentage for each age group last year.

There are now 6, prisoners. Originally, there were more in the 30—39 age group than the 20—29 age group. Determine the prison population for each age group last year. For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in Figure.

The total number of carbohydrates is g, and the total amount of fat is If there are more pieces of cashews than cranberries, how many of each item is in the trail mix? If there is the same amount of almonds as cashews, how many of each item is in the trail mix?

If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix? For the following exercises, determine whether the ordered pair is a solution to the system of equations. For the following exercises, use substitution to solve the system of equations. For the following exercises, write a system of equations to solve each problem.

Solve the system of equations. A factory has a cost of production and a revenue function What is the break-even point? A performer charges where is the total number of attendees at a show. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point? For the following exercises, solve the system of three equations using substitution or addition. Three odd numbers sum up to The smaller is one-third the larger and the middle number is 16 less than the larger.

What are the three numbers? A local theatre sells out for their show. For the following exercises, perform the requested operations on the given matrices. For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution. For the following exercises, write the augmented matrix from the system of linear equations.

For the following exercises, solve the system of linear equations using Gaussian elimination. For the following exercises, find the solutions by computing the inverse of the matrix. Students were asked to bring their favorite fruit to class. A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. How many brownies and how many cookies were sold?

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists. If what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse? For the following exercises, use Gaussian elimination to solve the systems of equations.

For the following exercises, use the inverse of a matrix to solve the systems of equations. A factory producing cell phones has the following cost and revenue functions: and What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

In one day, three times as many children as adults attended. How many of each type of ticket was sold? Skip to content Systems of Equations and Inequalities. Know the properties of determinants. Consider a system of two linear equations in two variables. Figure 1. Understanding Properties of Determinants There are many properties of determinants. Properties of Determinants If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.

When two rows are interchanged, the determinant changes sign. If either two rows or two columns are identical, the determinant equals zero. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero. The determinant of an inverse matrix is the reciprocal of the determinant of the matrix If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.

Illustrating Properties of Determinants Illustrate each of the properties of determinants. Key Concepts The determinant for is See Figure. Solutions are See Figure. Add the three diagonal entries upper left to lower right and subtract the three diagonal entries lower left to upper right.

See Figure and Figure. Certain properties of determinants are useful for solving problems. For example: If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal. Section Exercises Verbal Explain why we can always evaluate the determinant of a square matrix. Algebraic For the following exercises, find the determinant. Technology For the following exercises, use the determinant function on a graphing utility.

Real-World Applications For the following exercises, create a system of linear equations to describe the behavior. One number is 20 less than the other. Review Exercises Systems of Linear Equations: Two Variables For the following exercises, determine whether the ordered pair is a solution to the system of equations. Systems of Linear Equations: Three Variables For the following exercises, solve the system of three equations using substitution or addition.

Systems of Nonlinear Equations and Inequalities: Two Variables For the following exercises, solve the system of nonlinear equations. Partial Fractions For the following exercises, decompose into partial fractions. Matrices and Matrix Operations For the following exercises, perform the requested operations on the given matrices. Solving Systems with Gaussian Elimination For the following exercises, write the system of linear equations from the augmented matrix.

Solving Systems with Inverses For the following exercises, find the inverse of the matrix. Practice Test Is the following ordered pair a solution to the system of equations? Rewrite the system of linear equations as an augmented matrix.

Rewrite the augmented matrix as a system of linear equations. Previous: Solving Systems with Inverses. Next: Introduction to Analytic Geometry. Highest score default Date modified newest first Date created oldest first. Multiplying a row by a real number has the effect of multiplying the determinant by the same real number.

Swapping two adjacent rows of a matrix has the effect of changing the sign of the determinant of the matrix. Proof of 3. So if the original determinant was say And I happen to swap rows for 3 times. Am I right? This material should be standard at a first year linear algebra course in any university. Secondly, look at: en. Show 2 more comments. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.

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It also comes also consider looking alphabetical order Attributes Learning Labs as. The Thunderbirds brought computer program product enhancements to Windows Server Operating System. All used vehicles identifying redundant paths 2 professional and step 5.One way to answer this question would be to calculate each determinant using the cofactor expansion along the rows or columns. Thus, we can use the property that the determinant of a triangular matrix is equal to the product of the entries on the main diagonal. Let us further explore examples where we need to find the determinants of triangular matrices. In some cases, the easy part will be identifying that a matrix is triangular and applying the property for determinants and the hard part will involve further calculations to reach the required answer.

Diagonal matrices are a special kind of triangular matrix, and we can recall that the determinant of such a matrix is found by taking the product of the entries on the main diagonal. For our final example, let us find the determinant of a matrix given in terms of three variables that we will have to find by finding the determinants of smaller matrices. Since we have been given several equations with determinants in them and three unknown variables, the most obvious thing to do would be to find these determinants and see whether this gives us any information about the variables, so let us do this.

We can simplify the calculation of this matrix by noticing that it is an upper triangular matrix, since the entries below the main diagonal are zero:. Therefore, the determinant will just be the product of the entries on the main diagonal, giving us. We should be aware that both positive and negative 36 are possibilities here. Nagwa uses cookies to ensure you get the best experience on our website.

Learn more about our Privacy Policy. The portal has been deactivated. Please contact your portal admin. In this explainer, we will learn how to find the determinant of a triangular matrix. Definition: Triangular Matrix If the entries below the main diagonal are zero, the matrix is an upper triangular matrix. If the entries above the main diagonal are zero, the matrix is a lower triangular matrix. Upper and lower triangular matrices are shown:. Property: Determinants of Triangular Matrices The determinant of a triangular matrix is the product of the entries on the main diagonal:.

Answer One way to answer this question would be to calculate each determinant using the cofactor expansion along the rows or columns. Answer Since we have been given several equations with determinants in them and three unknown variables, the most obvious thing to do would be to find these determinants and see whether this gives us any information about the variables, so let us do this.

Key Points We can simplify the calculation of determinants in certain cases if some of the entries are zero. The first result concerns the determinant of a triangular matrix. Proposition Let be a triangular matrix either upper or lower. Then, the determinant of is equal to the product of its diagonal entries:. Suppose that is lower triangular. Denote by the set of all permutations of the first natural numbers. Let be the permutation in which the numbers are sorted in increasing order.

The parity of is even and its sign is because it does not contain any inversion see the lecture on the sign of a permutation. Then, the determinant of is where in step we have used the fact that for all permutations except the product involves at least one entry above the main diagonal that is equal to zero.

The latter fact can be proved by contradiction. Suppose the product involves only elements on the main diagonal or below it, and at least one element below it otherwise. Then, for all , but the inequality must be strict for at least one. Suppose that the inequality is strict for. Then, we have for. In other words, the permutation must contain different natural numbers smaller than or equal to , which is clearly impossible. This ends the proof by contradiction. Thus, we have proved the proposition for lower triangular matrices.

The proof for upper triangular matrices is almost identical we just need to reverse the inequalities in the last step. Proposition Let be an identity matrix. The identity matrix is diagonal. Therefore, it is triangular and its determinant is equal to the product of its diagonal entries.

The latter are all equal to. As a consequence, the determinant of is equal to. Proposition Let be a square matrix and denote its transpose by. For any permutation , there is an inverse permutation such that for. If is obtained by performing a sequence of transpositions , then is obtained by performing the opposite transpositions in reverse order. Thus, the number of transpositions is the same and, as a consequence, we have that By using the concept of inverse permutation, the determinant of can be easily calculated as follows: where: in step we have used the definition of transpose ; in step we have set and, as a consequence,.

The following property, while pretty intuitive, is often used to prove other properties of the determinant. Proposition Let be a square matrix. If has a zero row i. This property can be proved by using the definition of determinant For every permutation , we have that because the product contains one entry from each row column , but one of the rows columns contains only zeros.

Then is invertible if and only if and it is singular if and only if. The matrix is row equivalent to a unique matrix in reduced row echelon form RREF. Since and are row equivalent, we have that where are elementary matrices. Moreover, by the properties of the determinants of elementary matrices , we have that But the determinant of an elementary matrix is different from zero. Therefore, where is a non-zero constant.

If is invertible, is the identity matrix and If is singular, has at least one zero row because the only square RREF matrix that has no zero rows is the identity matrix, and the latter is row equivalent only to non-singular matrices.