Generate Quiz 697D78D568C1A

Solution: A matrix of identities is produced.Both B and C must be inverses of A according to the conditions AB=BA=I and AC=CA=I. B and C are therefore equal.

The calculation above makes options b, c, and d incorrect.

If A has a row of zeros and is a square matrix of order 2, then its determinant is zero.

There are n rows and one column in a matrix of order n x 1. We refer to such a matrix as a column matrix.

The determinant of a matrix is zero if any row is made up entirely of zeros.

Solution: We convert rows into columns and columns into rows during the transpose process.

Solution: If both A and B are inverses of each other, then multiplying the matrix yields the identity matrix.

Due to the calculation above, options a, b, and d are all incorrect.

((-1)^{i+j}M_{ij}) is the cofactor of the element a_ij in a matrix, where M_ij is the element a_ij’s minor.

When at least one diagonal element is non-zero (cij ≠ 0 for i = j) and all off-diagonal elements are zero (cij = 0 for i ≠ j), the square matrix is said to be diagonal.

The identity matrix I (of the same order) multiplied by a square matrix A yields the matrix itself. AI = IA = A as a result.

The provided matrix is a row matrix since it contains one row and several columns.

d) AB-1=(B-1 A-1) is the response.

Solution: The inverse of each individual matrix equals the product of the inverses of two matrices.

The calculation above makes options a, b, and c incorrect.

B = CA^(-1) is true if AB = C, and B = A^(-1)C is also true.

A product raised to a power raises each of the factors enclosed in parenthesis to that power separately.

Response: d) Unmultiplyable.

Solution: In order to enable matrix multiplication.

The first matrix’s number of columns equals the second matrix’s number of rows.

Order of Y=m×n X=l×m order

The aforementioned rule does not apply to the YX product.

Because of the calculation above, options a, b, and c are all incorrect.

One row and m columns make up a matrix of order 1 x m. We refer to such a matrix as a row matrix.

The requirement for a trivial solution to the system of linear homogeneous homogenous equations is

Raising the determinant to the power of the matrix’s order yields |2A| if |A| = 5. This means that (|2A| = (2^4) times |A| = 16 times 5 = 80.

(A^T = -A) is a property of a skew-symmetric matrix A. A skew-symmetric matrix’s inverse is likewise skew-symmetric. Consequently, (A^{-1} = -A).

The matrix itself is the inverse of the inverse of a non-singular matrix. Consequently, (A^-1)^-1 = A.

The response is d(None).

Solution: The individual transpose of matrices in reverse order, or DtCt, is equal to the transpose of the product of two matrices.

Let’s examine why the other choices are wrong.

a) CtDt = DCt

b) CD-1 = D-1C-1

c) DCt = CtDt

Solution: A square matrix A is referred to as a symmetric matrix if A=At.

Let’s examine the reasons why others are wrong.

b) The matrix is transposed when rows are converted to columns and vice versa.

c) A matrix that is square is referred to as non-symmetric if, At

d) A singular matrix is one in which A=0 is the determinant.

The determinant of a square matrix is the additive inverse of the determinant of the original matrix if any two of its columns are switched.

Solution: The determinant that results from removing the row and column that contain a given element is known as the minor of that element in a matrix.

Why the other choices are wrong

The formula A = (–1)i+j M, where M is a minor of aij, defines the cofactor of an element aij, represented by Aij.

c) The matrix’s modulus is not referred to as its determinant. Since a matrix is simply a collection of numbers, its modulus has no significance.

The set of values for x, y, and z that simultaneously satisfy all three equations is the system’s solution set. In this instance, all three equations are satisfied by (-3,0,5).

This is a characteristic of a determinant’s cofactor-based expansion. The determinant of the matrix is equal to the sum of the products of the elements of one row (or column) by the cofactors of the corresponding elements of another row (or column).

If and only if a square matrix contains two identical rows (or columns), its determinant becomes zero. Thus, |A| = 0 if any two rows in matrix A are the same.

If and only if a square matrix’s determinant (|A|) is not equal to zero, it is non-singular.