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If rank of matrix when the 3*3 square matrix have detrminants are not zero the rank is 1 br zero we go 2*2 sqare matric abd then determinants are zero then rank of matric is 2 or the non zero rank is 2?
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If rank of matrix when the 3*3 square matrix have detrminants are not ...
Determinants and Rank of Matrices
Determinants and rank are important concepts in linear algebra that help us understand the properties and behavior of matrices. The determinant of a square matrix is a scalar value that provides information about the matrix's invertibility and the system of linear equations it represents. The rank of a matrix, on the other hand, represents the maximum number of linearly independent rows or columns in the matrix.
Determinants and Rank of a 3x3 Square Matrix
A 3x3 square matrix can be represented as:
A = [ a11 a12 a13 ]
[ a21 a22 a23 ]
[ a31 a32 a33 ]
To calculate the determinant of this matrix, we can use the formula:
det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)
If the determinant of this 3x3 matrix is non-zero, it implies that the three rows/columns of the matrix are linearly independent, and hence the rank of the matrix is 3.
Rank of a 2x2 Matrix
If the determinant of a 2x2 matrix is non-zero, it means that the two rows/columns are linearly independent, and the rank of the matrix is 2.
If the determinant of a 2x2 matrix is zero, it implies that the two rows/columns are linearly dependent, and the rank of the matrix is 1.
Determinants and Rank of a Matrix
In general, for any square matrix of size n x n, if the determinant is non-zero, the rank of the matrix is n. This means that all n rows/columns are linearly independent, and the matrix is said to be full rank.
If the determinant is zero, it indicates that the rows/columns are linearly dependent, and the rank of the matrix is less than n. In this case, we need to consider smaller submatrices to determine the rank.
For example, if we have a 4x4 matrix with a zero determinant, we can calculate the determinants of its 3x3 submatrices. If any of these determinants are non-zero, it means the rank is 3. If all the determinants of the 3x3 submatrices are zero, we can calculate the determinants of the 2x2 submatrices. If any of these determinants are non-zero, the rank is 2. And if all the determinants of the 2x2 submatrices are zero, the rank is 1.
In conclusion, the rank of a matrix depends on the determinants of its submatrices. If the determinant of a 3x3 square matrix is non-zero, the rank is 3. If the determinant of a 2x2 square matrix is non-zero, the rank is 2. If the determinant of a 2x2 square matrix is zero, the rank is 1.
Determinants and rank are important concepts in linear algebra that help us understand the properties and behavior of matrices. The determinant of a square matrix is a scalar value that provides information about the matrix's invertibility and the system of linear equations it represents. The rank of a matrix, on the other hand, represents the maximum number of linearly independent rows or columns in the matrix.
Determinants and Rank of a 3x3 Square Matrix
A 3x3 square matrix can be represented as:
A = [ a11 a12 a13 ]
[ a21 a22 a23 ]
[ a31 a32 a33 ]
To calculate the determinant of this matrix, we can use the formula:
det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)
If the determinant of this 3x3 matrix is non-zero, it implies that the three rows/columns of the matrix are linearly independent, and hence the rank of the matrix is 3.
Rank of a 2x2 Matrix
If the determinant of a 2x2 matrix is non-zero, it means that the two rows/columns are linearly independent, and the rank of the matrix is 2.
If the determinant of a 2x2 matrix is zero, it implies that the two rows/columns are linearly dependent, and the rank of the matrix is 1.
Determinants and Rank of a Matrix
In general, for any square matrix of size n x n, if the determinant is non-zero, the rank of the matrix is n. This means that all n rows/columns are linearly independent, and the matrix is said to be full rank.
If the determinant is zero, it indicates that the rows/columns are linearly dependent, and the rank of the matrix is less than n. In this case, we need to consider smaller submatrices to determine the rank.
For example, if we have a 4x4 matrix with a zero determinant, we can calculate the determinants of its 3x3 submatrices. If any of these determinants are non-zero, it means the rank is 3. If all the determinants of the 3x3 submatrices are zero, we can calculate the determinants of the 2x2 submatrices. If any of these determinants are non-zero, the rank is 2. And if all the determinants of the 2x2 submatrices are zero, the rank is 1.
In conclusion, the rank of a matrix depends on the determinants of its submatrices. If the determinant of a 3x3 square matrix is non-zero, the rank is 3. If the determinant of a 2x2 square matrix is non-zero, the rank is 2. If the determinant of a 2x2 square matrix is zero, the rank is 1.
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If rank of matrix when the 3*3 square matrix have detrminants are not zero the rank is 1 br zero we go 2*2 sqare matric abd then determinants are zero then rank of matric is 2 or the non zero rank is 2? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about If rank of matrix when the 3*3 square matrix have detrminants are not zero the rank is 1 br zero we go 2*2 sqare matric abd then determinants are zero then rank of matric is 2 or the non zero rank is 2? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If rank of matrix when the 3*3 square matrix have detrminants are not zero the rank is 1 br zero we go 2*2 sqare matric abd then determinants are zero then rank of matric is 2 or the non zero rank is 2?.
If rank of matrix when the 3*3 square matrix have detrminants are not zero the rank is 1 br zero we go 2*2 sqare matric abd then determinants are zero then rank of matric is 2 or the non zero rank is 2? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about If rank of matrix when the 3*3 square matrix have detrminants are not zero the rank is 1 br zero we go 2*2 sqare matric abd then determinants are zero then rank of matric is 2 or the non zero rank is 2? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If rank of matrix when the 3*3 square matrix have detrminants are not zero the rank is 1 br zero we go 2*2 sqare matric abd then determinants are zero then rank of matric is 2 or the non zero rank is 2?.
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