Question

The rank of 3x3 matrix whose
elements are all is​

Answer 1

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?It is great that you’ve already taken the first step towards moving away from rote learning by asking “How?”. Rote learning is go

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?It is great that you’ve already taken the first step towards moving away from rote learning by asking “How?”. Rote learning is goThere are 29 total {−1,1} matrices.

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?It is great that you’ve already taken the first step towards moving away from rote learning by asking “How?”. Rote learning is goThere are 29 total {−1,1} matrices.Any such 3×3 matrix can be thought of geometrically as representing 3 planes in 3 dimensional Cartesian coordinates: The rows represent the normal vectors of each plane.

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?It is great that you’ve already taken the first step towards moving away from rote learning by asking “How?”. Rote learning is goThere are 29 total {−1,1} matrices.Any such 3×3 matrix can be thought of geometrically as representing 3 planes in 3 dimensional Cartesian coordinates: The rows represent the normal vectors of each plane.The matrix has non-zero determinant if and only if the 3 planes intersect at a single point (the origin). The only way for this to happen is if no two of the three unit normal vectors point either parallel or in opposite directions.

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?It is great that you’ve already taken the first step towards moving away from rote learning by asking “How?”. Rote learning is goThere are 29 total {−1,1} matrices.Any such 3×3 matrix can be thought of geometrically as representing 3 planes in 3 dimensional Cartesian coordinates: The rows represent the normal vectors of each plane.The matrix has non-zero determinant if and only if the 3 planes intersect at a single point (the origin). The only way for this to happen is if no two of the three unit normal vectors point either parallel or in opposite directions.Any normal vector with components from {−1,1} can point in 8 different directions corresponding to the exact centre line of each octant defined by the x,y and z axes.

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?It is great that you’ve already taken the first step towards moving away from rote learning by asking “How?”. Rote learning is goThere are 29 total {−1,1} matrices.Any such 3×3 matrix can be thought of geometrically as representing 3 planes in 3 dimensional Cartesian coordinates: The rows represent the normal vectors of each plane.The matrix has non-zero determinant if and only if the 3 planes intersect at a single point (the origin). The only way for this to happen is if no two of the three unit normal vectors point either parallel or in opposite directions.Any normal vector with components from {−1,1} can point in 8 different directions corresponding to the exact centre line of each octant defined by the x,y and z axes.#({−1,1} matrices with non-zero determinant)=8×6×4.

What is the total number of 3×3 matrices, whose elements belong to the set {-1,1}, having its determinant equal to 0?How should I break the habit of rote learning?It is great that you’ve already taken the first step towards moving away from rote learning by asking “How?”. Rote learning is goThere are 29 total {−1,1} matrices.Any such 3×3 matrix can be thought of geometrically as representing 3 planes in 3 dimensional Cartesian coordinates: The rows represent the normal vectors of each plane.The matrix has non-zero determinant if and only if the 3 planes intersect at a single point (the origin). The only way for this to happen is if no two of the three unit normal vectors point either parallel or in opposite directions.Any normal vector with components from {−1,1} can point in 8 different directions corresponding to the exact centre line of each octant defined by the x,y and z axes.#({−1,1} matrices with non-zero determinant)=8×6×4.Therefore our answer is:

#({−1,1} matrices with zero determinant)=29−8×6×4=320.(Answer)

Answer 2

The rank of a 3x3 matrix whose elements are all 2 is 1.

Given,

3x3 matrix whose elements are all 2

To Find,

The rank =?

Solution,

We can solve the question as follows:

It is asked that we have to find the rank of a 3x3 matrix whose elements are all 2.

The maximum number of non-zero independent rows or columns in a matrix is called the rank of a matrix. The rank of a 3x3 matrix can be 1, 2, or 3. It is 0 for a null matrix.

Here, since all the elements are 2, the matrix only has one independent row. The other two rows are dependent.

Hence, the rank of the 3x3 matrix whose elements are all 2 is 1.

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