Question

If a and b are matrices then which of the following is true
1) a+b not equal to b+a
2) (a^t)^t=a
3) ab not equal to ba
4) all are true

Answer 1

Answer:

only option 3) is true

suppose a=2. and b=3

then 2*3 = 3*2

please mark me at brainliest

Answer 2

Answer:

Options (2) and (3) are true.

Step-by-step explanation:

It is given that A and B are matrices.

Let us consider the $2\times2$ general matrices of A and B as follows:

$A=\left[\begin{array}{cc}a&b\\c&d\\\end{array}\right]$

$B=\left[\begin{array}{cc}w&x\\y&z\\\end{array}\right]$

(1) A + B is not equal to B + A.

Compute A + B as follows:

$A+B=\left[\begin{array}{cc}a&b\\c&d\\\end{array}\right]+\left[\begin{array}{cc}w&x\\y&z\\\end{array}\right]$

$A+B=\left[\begin{array}{cc}a+w&b+x\\c+y&d+z\\\end{array}\right]$

Similarly,

Compute B + A as follows:

$B+A=\left[\begin{array}{cc}w&x\\y&z\\\end{array}\right]+\left[\begin{array}{cc}a&b\\c&d\\\end{array}\right]$

$B+A=\left[\begin{array}{cc}w+a&x+b\\y+c&z+d\\\end{array}\right]$

$B+A=\left[\begin{array}{cc}a+w&b+x\\c+y&d+z\\\end{array}\right]$

Notice that A + B = B + A

Thus, option (1) is not true.

(2) $(A^T)^T=A$

Compute the transpose of the matrix A as follows:

$A^T=\left[\begin{array}{cc}a&c\\b&d\\\end{array}\right]$

Again, take the transpose as follows:

$(A^T)^T=\left[\begin{array}{cc}a&b\\c&d\\\end{array}\right]$

$(A^T)^T=A$

Thus, option (2) is true.

(3) AB is not equal to BA.

Compute AB as follows:

$AB=\left[\begin{array}{cc}a&b\\c&d\\\end{array}\right]\left[\begin{array}{cc}w&x\\y&z\\\end{array}\right]$

$AB=\left[\begin{array}{cc}aw+by&ax+bz\\cw+dy&cx+dz\\\end{array}\right]$

Similarly,

Compute BA as follows:

$BA=\left[\begin{array}{cc}w&x\\y&z\\\end{array}\right]\left[\begin{array}{cc}a&b\\c&d\\\end{array}\right]$

$BA=\left[\begin{array}{cc}wa+xc&wb+xd\\ya+zc&xc+dz\\\end{array}\right]$

Clearly AB ≠ BA.

Thus, option (3) is true.

(4) All are true.

As option (1) is not true.

Thus, option (4) is not true.

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