Determinants and Minors Quiz

True (A)

sec^2(θ) - tan^2(θ) = 1

<p>a_{ij}</p> Signup and view all the answers

<p>$sec^2(θ) - tan^2(θ)$ (C)</p> Signup and view all the answers

<p>Minor = Determinant after removing a row and column Secant = Reciprocal of cosine Tangent = Ratio of sine to cosine Determinant = Value computed from the elements of a square matrix</p> Signup and view all the answers

<p>$M_{12} = a_{21}a_{33} - a_{31}a_{23}$</p> Signup and view all the answers

<p>The determinant of the coefficient matrix (D)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>x + 2x + 6x - 1 = 0</p> Signup and view all the answers

<p>x</p> Signup and view all the answers

<p>D = Determinant of the coefficient matrix Dx = Determinant with modified first column Dy = Determinant with modified second column Dz = Determinant with modified third column</p> Signup and view all the answers

<p>k = –1 (D)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>A single equation within the system (D)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>1</p> Signup and view all the answers

<p>φ</p> Signup and view all the answers

<p>D must not equal 0</p> Signup and view all the answers

<p>determinants</p> Signup and view all the answers

<p>A(3, −1), B(0, −3), C(12, 5) = Collinear P(3, −5), Q(6, 1), R(4, 2) = Not Collinear L(0, 1), M(2, −1), N(−4, 7) = Not Collinear</p> Signup and view all the answers

<p>This equation does not contribute to the system of equations. (A)</p> Signup and view all the answers

<p>It changes its sign (C)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>The value of the new determinant becomes k times the value of the original determinant.</p> Signup and view all the answers

<p>changes its sign</p> Signup and view all the answers

<p>D can be simplified to A1 (B)</p> Signup and view all the answers

<p>Property 1 = The value of the new determinant is k times the value of the original Property 2 = Interchanging rows changes the sign of the determinant Property 3 = Identical rows result in a determinant of zero Property 4 = The determinant remains unchanged under certain operations</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>This indicates that the determinant D has changed its sign due to row interchange.</p> Signup and view all the answers

<p>k times the value of the given determinant</p> Signup and view all the answers

<p>6 (A)</p> Signup and view all the answers

<p>True (A)</p> Signup and view all the answers

<p>x = -c2/b2 and y = -c3/b3</p> Signup and view all the answers

<p>-5</p> Signup and view all the answers

<p>x = 15 y = 12 z = 6 D = -5</p> Signup and view all the answers

<p>All of the above (D)</p> Signup and view all the answers

<p>False (B)</p> Signup and view all the answers

<p>x + y - 2 = 0, 2x + 3y - 5 = 0, 3x - 2y - 1 = 0</p> Signup and view all the answers

<p>[[1, 1], [2, 3], [3, -2]]</p> Signup and view all the answers

<p>-5 (B)</p> Signup and view all the answers

The value obtained by deleting the ith row and jth column of a matrix A and calculating the determinant of the remaining submatrix. For example, the minor of the element a11 is the determinant of the matrix obtained by deleting the first row and first column of A.

The value obtained by multiplying the minor of an element aij by (-1)^(i+j).

The determinant obtained by substituting any row or column of the original matrix with its corresponding cofactors.

A method for calculating the determinant of a square matrix by expanding along a row or column using cofactors.

A process used to calculate the inverse of a matrix, where the adjoint matrix is then divided by the determinant.

The determinant of a matrix is non-zero.

The determinant of a matrix equals zero.

Swapping two rows of a determinant changes its sign.

If two rows of a determinant are identical, the determinant's value is zero.

Multiplying a row of a determinant by a constant 'k' multiplies the determinant's value by 'k'.

The process of adding a multiple of one row to another row in a determinant.

A determinant with two rows or columns interchanged has its sign flipped.

If a determinant has two identical rows or columns, its value is zero.

Multiplying every element of a row (or column) of a determinant by a constant 'k' multiplies the determinant's value by 'k'.

The method of expressing a determinant by expanding along a row or column.

A set of operations applied to a determinant are performed on a single row or column at a time.

A matrix is non-singular if its determinant is non-zero. This means the matrix has an inverse and can be used to solve linear equations.

A matrix is singular if its determinant equals zero. This means the matrix does not have an inverse and cannot be used to solve linear equations.

The determinant of a square matrix is a numerical value that represents certain properties of the matrix, such as its invertibility. It can be calculated using various methods like cofactor expansion.

The cofactor of an element in a matrix is calculated by multiplying the minor of that element by (-1) raised to the power of the sum of its row and column indices. It plays a key role in finding the determinant.

Laplace's theorem states that the determinant of a matrix can be calculated by expanding along any row or column using the cofactors of the elements in that row or column. It provides a method for finding the determinant.

A system of equations is a collection of two or more equations that share the same variables and must be solved simultaneously.

Cramer's rule is a method used to solve systems of linear equations by using determinants. It involves calculating determinants for the main coefficient matrix (D), and for matrices where the constant terms replace the coefficients of each variable (Dx, Dy, Dz).

A determinant is a scalar value calculated from the elements of a square matrix. It represents certain properties of the matrix.

The determinant 'D' is calculated from the coefficients of the variables in the system of equations. If D is not equal to zero, there is a unique solution to the system.

When solving for a specific variable (e.g., x), the 'Dx' determinant is calculated by replacing the column of coefficients for that variable (e.g., 'x') with the constants from the right side of the equations.

The solution to the system of equations is found by dividing the determinants 'Dx', 'Dy', 'Dz' by the determinant 'D', where each determinant represents the solution for the respective variable.

If the determinant 'D' is equal to zero, the Cramer's rule cannot be applied, and there is no unique solution to the system of equations. This means there might be infinitely many solutions or no solution at all.

The coefficients of the variables in the system of equations are arranged in a matrix form where the first row represents the first equation, the second row represents the second equation, and so on.

A system of equations is considered linear if the variables are only raised to the power of 1. The equations form straight lines when graphed.

To solve a system of linear equations, you need to find the values of the variables that satisfy all equations simultaneously.

A system of equations is considered consistent if it has at least one solution. This means the equations have at least one point in common where they intersect.

The determinant of a matrix (a square array of numbers) is a specific value calculated from its elements. It helps determine if a matrix is singular or non-singular.

The adjoint of a matrix is a special matrix derived from a given matrix by taking the transpose of the matrix of its cofactors.

A cofactor of an element in a matrix is found by multiplying the minor of that element by (-1)^(i+j), where ‘i’ and ‘j’ are its row and column positions.

The minor of an element in a matrix is the determinant of a sub-matrix formed by removing the row and column containing that element.

The Inverse of a Matrix is a matrix that when multiplied with the original matrix results in an identity matrix. It's like the 'opposite' or 'undo' operation.