Let e be the error in b. Assuming that A is a square matrix, the error in the solution A < sup >− 1 </ sup > b is A < sup >− 1 </ sup > e.

In linear algebra, the determinant is a value associated with a square matrix.

A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.

There are various ways to define the determinant of a square matrix A, i. e. one with the same number of rows and columns.

where b and c are scalars, v is any vector of size n and I is the identity matrix of size n. These properties state that the determinant is an alternating multilinear function of the columns, and they suffice to uniquely calculate the determinant of any square matrix.

Assume is a square matrix with rows and columns, so that it can be written as

It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix.

Another equivalent definition of the rank of a matrix is the greatest order of any non-zero minor in the matrix ( the order of a minor being the size of the square sub-matrix of which it is the determinant ).

The rank of a square matrix can also be characterized as the tensor rank: the minimum number of simple tensors ( rank 1 tensors ) needed to express A as a linear combination,.

* If A is a square matrix ( i. e., m = n ), then A is invertible if and only if A has rank n ( that is, A has full rank ).

Their team of architect Bruce Graham and structural engineer Fazlur Khan designed the building as nine square " tubes ", each essentially a separate building, clustered in a 3 × 3 matrix forming a square base with 225-foot ( 75 m ) sides.

This variance is a positive semi-definite square matrix, commonly referred to as the covariance matrix.

This variance is also a positive semi-definite square matrix.

* Skew-symmetric matrix, a square matrix whose transpose is also its negative

* Skew-Hermitian matrix, a complex square matrix whose conjugate transpose is also its negative

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal ( the diagonal from the upper left to the lower right ) of A, i. e.,

Note that the trace is only defined for a square matrix ( i. e., n × n ).

It is remembered and has been commemorated by a bust in a park and a square in the city which was renamed Piazzo Lauro Di Bosis after the war.

The square has one corner point on the straight line segment, and does not lie entirely in the interior.

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.

If the square mile is divided into quarters, each quarter has a side length of mile ( 880 yards ) and is square mile in area, or 160 acres.

As a unit of measure, an acre has no prescribed shape ; any perimeter enclosing 43, 560 square feet is an acre in size.

* 0. 40468564224 hectare ( A square with 100 m sides has an area of 1 hectare.

As of late 2005, the area once devoted to a live-action facsimile of Dogpatch ( including a lifesize statue in the town square of Dogpatch " founder ," General Jubilation T. Cornpone ) has been heavily stripped by vandals and souvenir hunters, and is today slowly being reclaimed by the surrounding Arkansas wilderness.

Borneo has 19, 800, 000 inhabitants ( in mid 2010 ), a population density of 26 inhabitants per square km.

Castling consists of moving the king two squares along the first rank toward a rook ( which is on the player's first rank ) and then placing the rook on the last square the king has just crossed.

O ' Flaherty then asks that we suppose that the area to be protected is square and each hectare inside it has the same value of protection.

Nzwani, triangular shaped and forty kilometers from apex to base, has an area of 424 square kilometers.

It has a density of 77 people per square kilometer.

A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square.

The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square.

The prototype Ob. 9 had a commander's cupola with DShK 1938 / 46 machine gun, but the production model has a square commander's hatch, opening to the right.

It rises from the stylobate without any base ; it is from four to six times as tall as its diameter ; it has twenty broad flutes ; the capital consists simply of a banded necking swelling out into a smooth echinus, which carries a flat square abacus ; the Doric entablature is also the heaviest, being about one-fourth the height column.

This implies that square wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of inverted catenary curves.

Normally, the software in a computer treats the computer ’ s display screen as a rectangular array of square, indivisible pixels, each of which has an intensity and color that are determined by the blending of three primary colors: red, green, and blue.

Delphi also has a school, a lyceum, a church and a square ( plateia ).

Over his career he mapped over 3. 9 million square kilometers of North America and for this has been described as the " greatest land geographer who ever lived.

It has been estimated that some 10 – 20 % of drylands are already degraded, the total area affected by desertification being between 6 and 12 million square kilometres, that about 1 – 6 % of the inhabitants of drylands live in desertified areas, and that a billion people are under threat from further desertification.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

Dublin has more green spaces per square kilometre than any other European capital city, with 97 % of city residents living within 300 metres of a park area.

He also lacked a symbol for a general number n. Where we would write, Diophantus has to resort to constructions like: ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.