In one sentence, describe what the following article is about: Remember, you can pick any row or column as your reference. You'll get the same answer no matter which you pick. If you pick a row or column with zeros, you only need to calculate the cofactor for the nonzero elements. Here's why: Let's say you pick row 2, with elements a21, a22, and a23. To solve this problem, we'll be looking at three different 2x2 matrices. Let's call them A21, A22, and A23. The determinant of the 3x3 matrix is a21|A21| - a22|A22| + a23|A23|. If terms a22 and a23 are both 0, our formula becomes a21|A21| - 0*|A22| + 0*|A23| = a21|A21| - 0 + 0 = a21|A21|. Now we only have to calculate the cofactor of a single element. If you take the values of one row and add them to a different row, the determinant of the matrix does not change. The same is true of columns. You can do this repeatedly or multiply the values by a constant before adding to get as many zeroes in the matrix as possible. This can save you a lot of time. For example, say you have a 3 x 3 matrix: (912310752){\displaystyle {egin{pmatrix}9&-1&2\3&1&0\7&5&-2\end{pmatrix}}} In order to cancel out the 9 in position a11, we can multiply the second row by -3 and add the result to the first. The new first row is [9 -1 2] + [-9 -3 0] = [0 -4 2]. The new matrix is (042310752){\displaystyle {egin{pmatrix}0&-4&2\3&1&0\7&5&-2\end{pmatrix}}} Try to use the same trick with columns to turn a12 into a 0 as well. In these special cases, the determinant is simply the product of the elements along the main diagonal, from a11 in the top left to a33 in the lower right. We're still talking about 3x3 matrices, but "triangular" ones have special patterns of nonzero values: Upper triangular matrix: All the non-zero elements are on or above the main diagonal. Everything below is a zero. Lower triangular matrix: All the non-zero elements are on or below the main diagonal. Diagonal matrix: All the non-zero elements are on the main diagonal. (A subset of the above.)
Summary: Pick the reference with the most zeroes. Use row addition to make the matrix easier. Learn the shortcut for triangular matrices.

In one sentence, describe what the following article is about: One symptom of keratitis is red, swollen eyes. Swollen eyes may be accompanied by a discharge from the eye. The discharge from the eye may be clear and watery, making the fur beneath look wet. It may also take on a yellow-green color if it is infected. Swollen eyes may be a symptom of a generalized infection from the herpes virus. Cats with keratitis may display symptoms of eye discomfort. This means the cat may favor the eye. Your cat may rub the eye with his paw or rub his eye on the ground. The eye may be closed, or your cat may start squinting. Additionally, he may be sensitive to light, so he may spend time in dark places or have trouble keeping his eyes open in bright lights. If your cat has keratitis, there may be some discoloration in the eye. The surface in the center of the eye should be clear so you can see the black pupil. If keratitis is present, you may see abnormalities there. The surface of the eye may have inflamed patches with blood vessels threading over the surface. Sometimes, due to extra cells in the eye area, the eye may take on a whitish appearance that looks like white-pink lace. With severe keratitis that becomes well-established due to lack of treatment, sometimes the area slowly becomes pigmented and brown staining covers the surface. This change may be permanent and the pigment prevents light entering the eye and interferes with the cat's sight. In severe cases, the cat may become blind in that eye as a result. Therefore, early diagnosis and treatment is desirable. Tests may be done to rule out other possible causes of the eye symptoms to make a more definitive diagnosis, especially in the case of herpes. Cats who have keratitis may display signs pointing to problems with their upper respiratory system. This might include sneezing, nasal discharge, or even discharge from the ears. If the keratitis occurs because of the herpes virus, respiratory symptoms may show earlier than or at the same time as eye symptoms.
Summary: Watch for swollen eyes. Check to see if the cat is favoring his eye. Monitor for eye discoloration. Look for brown stains in the eyes. Look for respiratory distress.

In one sentence, describe what the following article is about: Think about what interests you, and learn a skill in that field. Adding more skills to your repertoire will boost your confidence and provide more opportunities for career growth. If you haven't revisited your resume in a while, open it up and give it a refresh. Add any new skills and experiences you have had. Take the time to ensure that it is professional looking. Take a look at our guide on writing resumes for helpful tips. Take your career in a different direction. Look for opportunities within the company for different positions, and if that fails then look for a new company. If you're finding that you just can't advance in your career, then it may be time to go back to school. Getting the next level degree in your field will add to your marketability, and going to school can provide a dramatic scenery change that may inspire you to change paths completely.
Summary:
Learn a new skill. Write a new resume. Make a change. Go back to school.